Chapter 8 Resource Masters - Morgan Park High School · 2010. 5. 3. · © Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide to Using the Chapter 8 Resource Masters The Fast

Chapter 8Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks in both English and Spanish.

Study Guide and Intervention Workbook 0-07-827753-1Study Guide and Intervention Workbook (Spanish) 0-07-827754-XSkills Practice Workbook 0-07-827747-7Skills Practice Workbook (Spanish) 0-07-827749-3Practice Workbook 0-07-827748-5Practice Workbook (Spanish) 0-07-827750-7

ANSWERS FOR WORKBOOKS The answers for Chapter 8 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 1. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-827732-9 Algebra 1Chapter 8 Resource Masters

2 3 4 5 6 7 8 9 10 024 11 10 09 08 07 06 05 04 03

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 1

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 8-1Study Guide and Intervention . . . . . . . . 455–456Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 457Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 458Reading to Learn Mathematics . . . . . . . . . . 459Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 460




Lesson 8-5Study Guide and Intervention . . . . . . . 479–480Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 481Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Reading to Learn Mathematics . . . . . . . . . . 483Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 484




Chapter 8 AssessmentChapter 8 Test, Form 1 . . . . . . . . . . . . 503–504Chapter 8 Test, Form 2A . . . . . . . . . . . 505–506Chapter 8 Test, Form 2B . . . . . . . . . . . 507–508Chapter 8 Test, Form 2C . . . . . . . . . . . 509–510Chapter 8 Test, Form 2D . . . . . . . . . . . 511–512Chapter 8 Test, Form 3 . . . . . . . . . . . . 513–514Chapter 8 Open-Ended Assessment . . . . . . 515Chapter 8 Vocabulary Test/Review . . . . . . . 516Chapter 8 Quizzes 1 & 2 . . . . . . . . . . . . . . . 517Chapter 8 Quizzes 3 & 4 . . . . . . . . . . . . . . . 518Chapter 8 Mid-Chapter Test . . . . . . . . . . . . 519Chapter 8 Cumulative Review . . . . . . . . . . . 520Chapter 8 Standardized Test Practice . . 521–522

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A35

© Glencoe/McGraw-Hill iv Glencoe Algebra 1

Teacher’s Guide to Using theChapter 8 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 8 Resource Masters includes the core materials neededfor Chapter 8. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 1 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 8-1.Encourage them to add these pages to theirAlgebra Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 1 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 1

Assessment OptionsThe assessment masters in the Chapter 8Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 1. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 470–471. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

88

© Glencoe/McGraw-Hill vii Glencoe Algebra 1

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 8.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

binomial

by·NOH·mee·uhl

constant

degree of a monomial

degree of a polynomial

FOIL method

monomial

mah·NOH·mee·uhl

negative exponent

polynomial

PAH·luh·NOH·mee·uhl

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 1

Vocabulary Term Found on Page Definition/Description/Example

Power of a Power

Power of a Product

Product of Powers

Power of a Quotient

Quotient of Powers

scientific notation

trinomial

try·NOH·mee·uhl

zero exponent

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

88

Study Guide and InterventionMultiplying Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

6-18-1

© Glencoe/McGraw-Hill 455 Glencoe Algebra 1

Less

on

8-1

Multiply Monomials A monomial is a number, a variable, or a product of a numberand one or more variables. An expression of the form xn is called a power and representsthe product you obtain when x is used as a factor n times. To multiply two powers that havethe same base, add the exponents.

Product of Powers For any number a and all integers m and n, am � an � am � n.

Simplify (3x6)(5x2).

(3x6)(5x2) � (3)(5)(x6 � x2) Associative Property

� (3 � 5)(x6 � 2) Product of Powers

� 15x8 Simplify.

The product is 15x8.

Simplify (�4a3b)(3a2b5).

(�4a3b)(3a2b5) � (�4)(3)(a3 � a2)(b � b5)� �12(a3 � 2)(b1 � 5)� �12a5b6

The product is �12a5b6.

Example 1Example 1 Example 2Example 2

ExercisesExercises

Simplify.

1. y( y5) 2. n2 � n7 3. (�7x2)(x4)

4. x(x2)(x4) 5. m � m5 6. (�x3)(�x4)

7. (2a2)(8a) 8. (rs)(rs3)(s2) 9. (x2y)(4xy3)

10. (2a3b)(6b3) 11. (�4x3)(�5x7) 12. (�3j2k4)(2jk6)

13. (5a2bc3)� abc4� 14. (�5xy)(4x2)( y4) 15. (10x3yz2)(�2xy5z)1�5

1�3


Powers of Monomials An expression of the form (xm)n is called a power of a powerand represents the product you obtain when xm is used as a factor n times. To find thepower of a power, multiply exponents.

Power of a Power For any number a and all integers m and n, (am)n � amn.

Power of a Product For any number a and all integers m and n, (ab)m � ambm.

Simplify (�2ab2)3(a2)4.

(�2ab2)3(a2)4 � (�2ab2)3(a8) Power of a Power

� (�2)3(a3)(b2)3(a8) Power of a Product

� (�2)3(a3)(a8)(b2)3 Commutative Property

� (�2)3(a11)(b2)3 Product of Powers

� �8a11b6 Power of a Power

The product is �8a11b6.

Simplify.

1. (y5)2 2. (n7)4 3. (x2)5(x3)

4. �3(ab4)3 5. (�3ab4)3 6. (4x2b)3

7. (4a2)2(b3) 8. (4x)2(b3) 9. (x2y4)5

10. (2a3b2)(b3)2 11. (�4xy)3(�2x2)3 12. (�3j2k3)2(2j2k)3

13. (25a2b)3� abc�214. (2xy)2(�3x2)(4y4) 15. (2x3y2z2)3(x2z)4

16. (�2n6y5)(�6n3y2)(ny)3 17. (�3a3n4)(�3a3n)4 18. �3(2x)4(4x5y)2

1�5

Study Guide and Intervention (continued)

Multiplying Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

ExampleExample

ExercisesExercises

Skills PracticeMultiplying Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1


Less

on

8-1

Determine whether each expression is a monomial. Write yes or no. Explain.

1. 11

2. a � b

3.

4. y

5. j3k

6. 2a � 3b

Simplify.

7. a2(a3)(a6) 8. x(x2)(x7)

9. (y2z)(yz2) 10. (�2k2)(�3k)

11. (e2f4)(e2f 2) 12. (cd2)(c3d2)

13. (2x2)(3x5) 14. (5a7)(4a2)

15. (4xy3)(3x3y5) 16. (7a5b2)(a2b3)

17. (�5m3)(3m8) 18. (�2c4d)(�4cd)

19. (102)3 20. (p3)12

21. (�6p)2 22. (�3y)3

23. (3pq2)2 24. (2b3c4)2

GEOMETRY Express the area of each figure as a monomial.

25. 26. 27.

4p

9p3cd

cdx2

x5

p2�q2


Determine whether each expression is a monomial. Write yes or no. Explain.

1.

2.

Simplify.

3. (�5x2y)(3x4) 4. (2ab2c2)(4a3b2c2)

5. (3cd4)(�2c2) 6. (4g3h)(�2g5)

7. (�15xy4)�� xy3� 8. (�xy)3(xz)

9. (�18m2n)2�� mn2� 10. (0.2a2b3)2

11. � p�212. � cd3�2

13. (0.4k3)3 14. [(42)2]2

GEOMETRY Express the area of each figure as a monomial.

15. 16. 17.

GEOMETRY Express the volume of each solid as a monomial.

18. 19. 20.

21. COUNTING A panel of four light switches can be set in 24 ways. A panel of five lightswitches can set in twice this many ways. In how many ways can five light switches be set?

22. HOBBIES Tawa wants to increase her rock collection by a power of three this year andthen increase it again by a power of two next year. If she has 2 rocks now, how manyrocks will she have after the second year?

7g2

3g

m3nmn3

n

3h23h2

3h2

6ac3

4a2c

5x3

6a2b4

3ab2

1�4

2�3

1�6

1�3

b3c2�2

21a2�7b

Practice Multiplying Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Reading to Learn MathematicsMultiplying Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1


Less

on

8-1

Pre-Activity Why does doubling speed quadruple braking distance?

Read the introduction to Lesson 8-1 at the top of page 410 in your textbook.

Find two examples in the table to verify the statement that when speed isdoubled, the braking distance is quadrupled. Write your examples in thetable.

Speed Braking Distance Speed Doubled Braking Distance (miles per hour) (feet) (miles per hour) Quadrupled (feet)

Reading the Lesson

1. Describe the expression 3xy using the terms monomial, constant, variable, and product.

2. Complete the chart by choosing the property that can be used to simplify eachexpression. Then simplify the expression.

Expression Property Expression Simplified

Product of Powers

35 � 32 Power of a Power

Power of a Product

Product of Powers

(a3)4 Power of a Power

Power of a Product

Product of Powers

(�4xy)5 Power of a Power

Power of a Product

Helping You Remember

3. Write an example of each of the three properties of powers discussed in this lesson.Then, using the examples, explain how the property is used to simplify them.


An Wang An Wang (1920–1990) was an Asian-American who became one of thepioneers of the computer industry in the United States. He grew up inShanghai, China, but came to the United States to further his studiesin science. In 1948, he invented a magnetic pulse controlling devicethat vastly increased the storage capacity of computers. He laterfounded his own company, Wang Laboratories, and became a leader inthe development of desktop calculators and word processing systems.In 1988, Wang was elected to the National Inventors Hall of Fame.

Digital computers store information as numbers. Because theelectronic circuits of a computer can exist in only one of two states,open or closed, the numbers that are stored can consist of only twodigits, 0 or 1. Numbers written using only these two digits are calledbinary numbers. To find the decimal value of a binary number, youuse the digits to write a polynomial in 2. For instance, this is how tofind the decimal value of the number 10011012. (The subscript 2 indicates that this is a binary number.)

10011012 � 1 � 2�

6�

� 0 � 2�

5�

� 0 � 2�

4�

� 1 � 2�

3�

� 1 � 2�

2�

� 0 � 2�

1�

� 1 � 2�

0�

� 1 � 6�4�

� 0 � 3�2�

� 0 � 1�6�

� 1 � 8�

� 1 � 4�

� 0 � 2�

� 1 � 1�

� 64 � 0 � 0 � 8 � 4 � 0 � 1

� 77

Find the decimal value of each binary number.

1. 11112 2. 100002 3. 110000112 4. 101110012

Write each decimal number as a binary number.

5. 8 6. 11 7. 29 8. 117

9. The chart at the right shows a set of decimal code numbers that is used widely in storing letters of the alphabet in a computer's memory.Find the code numbers for the letters of your name. Then write the code for your name using binary numbers.

The American Standard Guide for Information Interchange (ASCII)

A 65 N 78 a 97 n 110B 66 O 79 b 98 o 111C 67 P 80 c 99 p 112D 68 Q 81 d 100 q 113E 69 R 82 e 101 r 114F 70 S 83 f 102 s 115G 71 T 84 g 103 t 116H 72 U 85 h 104 u 117I 73 V 86 i 105 v 118J 74 W 87 j 106 w 119K 75 X 88 k 107 x 120L 76 Y 89 l 108 y 121M 77 Z 90 m 109 z 122

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Study Guide and InterventionDividing Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

6-28-2


Less

on

8-2

Quotients of Monomials To divide two powers with the same base, subtract theexponents.

Quotient of Powers For all integers m and n and any nonzero number a, � am � n.

Power of a Quotient For any integer m and any real numbers a and b, b � 0, � �m� .am

�bm

a�b

am�an

Simplify . Assume

neither a nor b is equal to zero.

� � �� Group powers with the same base.

� (a4 � 1)(b7 � 2) Quotient of Powers

� a3b5 Simplify.

The quotient is a3b5 .

b7�b2

a4�a

a4b7�ab2

a4b7�ab2 Simplify � �3

.

Assume that b is not equal to zero.

� �3� Power of a Quotient

� Power of a Product

� Power of a Power

� Quotient of Powers

The quotient is .8a9b9�27

8a9b9�27

8a9b15�27b6

23(a3)3(b5)3��

(3)3(b2)3

(2a3b5)3�

(3b2)32a3b5�

3b2

2a3b5�

3b2Example 1Example 1 Example 2Example 2

ExercisesExercises

Simplify. Assume that no denominator is equal to zero.

1. 2. 3.

4. 5. 6.

7. 8. � �39. � �3

10. � �411. � �4

12. r7s7t2�s3r3t2

3r6s3�2r5s

2v5w3�v4w3

4p4q4�3p2q2

2a2b�a

xy6�y4x

�2y7�14y5

x5y3�x5y2

a2�a

p5n4�p2n

m6�m4

55�52


Negative Exponents Any nonzero number raised to the zero power is 1; for example,(�0.5)0 � 1. Any nonzero number raised to a negative power is equal to the reciprocal of the number raised to the opposite power; for example, 6�3 � . These definitions can be used to simplify expressions that have negative exponents.

Zero Exponent For any nonzero number a, a0 � 1.

Negative Exponent Property For any nonzero number a and any integer n, a�n � and � an.

The simplified form of an expression containing negative exponents must contain onlypositive exponents.

Simplify . Assume that the denominator is not equal to zero.

� � �� Group powers with the same base.

� (a�3 � 2)(b6 � 6)(c5) Quotient of Powers and Negative Exponent Properties

� a�5b0c5 Simplify.

� � �(1)c5 Negative Exponent and Zero Exponent Properties

� Simplify.

The solution is .


1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. � �0 12.(�2mn2)�3��

4m�6n44m2n2�8m�1�

s�3t�5�(s2t3)�1

(3st)2u�4��s�1t2u7

(6a�1b)2��

(b2)4

x4y0�x�2

(a2b3)2�(ab)�2

(�x�1y)0��4w�1y2

b�4�b�5

p�8�p3

m�m�4

22�2�3

c5�4a5

c5�4a5

1�a5

1�4

1�4

1�4

1�c�5

b6�b6

a�3�a2

4�16

4a�3b6��16a2b6c�5

4a�3b6��16a2b6c�5

1�a�n

1�an

1�63


Dividing Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

ExampleExample

ExercisesExercises

Skills PracticeDividing Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


Less

on

8-2


1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. � �214. 4�4

15. 8�2 16. � ��2

17. � ��118.

19. k0(k4)(k�6) 20. k�1(��6)(m3)

21. 22. � �0

23. 24.

25. 26. 48x6y7z5��6xy5z6

�15w0u�1��

5u3

15x6y�9�5xy�11

f�5g4�h�2

16p5q2�2p3q3

f�7�f4

h3�h�6

9�11

5�3

4p7�7s2

32x3y2z5��8xyz2

�21w5u2��

7w4u5

m7n2�m3n2

a3b5�ab2

w4u3�w4u

12n5�36n

9d7�3d6

m�m3

r3s2�r3s4

x4�x2

912�98

65�64



1. 2. 3.

4. 5. 6.

7. � �38. � �2

9.

10. x3( y�5)(x�8) 11. p(q�2)(r�3) 12. 12�2

13. � ��214. � ��4

15.

16. 17. 18. � �0

19. 20. 21.

22. 23. 24.

25. � ��526. � ��1

27. � ��2

28. BIOLOGY A lab technician draws a sample of blood. A cubic millimeter of the bloodcontains 223 white blood cells and 225 red blood cells. What is the ratio of white bloodcells to red blood cells?

29. COUNTING The number of three-letter “words” that can be formed with the Englishalphabet is 263. The number of five-letter “words” that can be formed is 265. How manytimes more five-letter “words” can be formed than three-letter “words”?

2x3y2z�3x4yz�2

7c�3d3�c5de�4

q�1r3�qr�2

(2a�2b)�3��

5a2b4( j�1k3)�4�

j3k3

m�2n�5��(m4n3)�1

r4�(3r)3

�12t�1u5v�4��

2t�3uv56f�2g3h5��54f�2g�5h3

x�3y5�4�3

8c3d2f4��4c�1d2f�3

�15w0u�1��

5u3

22r3s2�11r2s�3

4�3

3�7

�4c2�24c5

6w5�7p6s3

4f 3g�3h6

8y7z6�4y6z5

5c2d3��4c2d

m5np�m4p

xy2�xy

a4b6�ab3

88�84

Practice Dividing Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

Reading to Learn MathematicsDividing Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


Less

on

8-2

Pre-Activity How can you compare pH levels?


• In the formula c � � �pH, identify the base and the exponent.

• How do you think c will change as the exponent increases?

Reading the Lesson

1. Explain what the statement � am � n means.

2. To find c in the formula c � � �pH, you can find the power of the numerator, the power of

the denominator, and divide. This is an example of what property?

3. Use the Quotient of Powers Property to explain why 30 � 1.

4. Consider the expression 4�3.

a. Explain why the expression 4�3 is not simplified.

b. Define the term reciprocal.

c. 4�3 is the reciprocal of what power of 4?

d. What is the simplified form of 4�3?


5. Describe how you would help a friend who needs to simplify the expression .4x2�2x5

1�10

am�an

1�10


Patterns with PowersUse your calculator, if necessary, to complete each pattern.

a. 210 � b. 510 � c. 410 �

29 � 59 � 49 �

28 � 58 � 48 �

27 � 57 � 47 �

26 � 56 � 46 �

25 � 55 � 45 �

24 � 54 � 44 �

23 � 53 � 43 �

22 � 52 � 42 �

21 � 51 � 41 �

Study the patterns for a, b, and c above. Then answer the questions.

1. Describe the pattern of the exponents from the top of each column to the bottom.

2. Describe the pattern of the powers from the top of the column to the bottom.

3. What would you expect the following powers to be?20 50 40

4. Refer to Exercise 3. Write a rule. Test it on patterns that you obtain using 22, 25, and 24as bases.

Study the pattern below. Then answer the questions.

03 � 0 02 � 0 01 � 0 00 � 0�1 does not exist. 0�2 does not exist. 0�3 does not exist.

5. Why do 0�1, 0�2, and 0�3 not exist?

6. Based upon the pattern, can you determine whether 00 exists?

7. The symbol 00 is called an indeterminate, which means that it has no unique value.Thus it does not exist as a unique real number. Why do you think that 00 cannot equal 1?

?

Enrichment

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Study Guide and InterventionScientific Notation

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Scientific Notation Keeping track of place value in very large or very small numberswritten in standard form may be difficult. It is more efficient to write such numbers inscientific notation. A number is expressed in scientific notation when it is written as aproduct of two factors, one factor that is greater than or equal to 1 and less than 10 and onefactor that is a power of ten.

Scientific NotationA number is in scientific notation when it is in the form a � 10n, where 1 � a � 10 and n is an integer.

Express 3.52 � 104 instandard notation.

3.52 � 104 � 3.52 � 10,000� 35,200

The decimal point moved 4 places to theright.

Express 37,600,000 inscientific notation.

37,600,000 � 3.76 � 107

The decimal point moved 7 places so that itis between the 3 and the 7. Since 37,600,000 � 1, the exponent is positive.

Express 6.21 � 10�5 instandard notation.

6.21 � 10�5 � 6.21 �

� 6.21 � 0.00001� 0.0000621

The decimal point moved 5 places to the left.

Express 0.0000549 inscientific notation.

0.0000549 � 5.49 � 10�5

The decimal point moved 5 places so that itis between the 5 and the 4. Since 0.0000549 � 1, the exponent is negative.

1�105



ExercisesExercises

Express each number in standard notation.

1. 3.65 � 105 2. 7.02 � 10�4 3. 8.003 � 108

4. 7.451 � 106 5. 5.91 � 100 6. 7.99 � 10�1

7. 8.9354 � 1010 8. 8.1 � 10�9 9. 4 � 1015

Express each number in scientific notation.

10. 0.0000456 11. 0.00001 12. 590,000,000

13. 0.00000000012 14. 0.000080436 15. 0.03621

16. 433 � 104 17. 0.0042 � 10�3 18. 50,000,000,000


Products and Quotients with Scientific Notation You can use properties ofpowers to compute with numbers written in scientific notation.

Evaluate (6.7 � 103)(2 � 10�5). Express the result in scientific andstandard notation.

(6.7 � 103)(2 � 10�5) � (6.7 � 2)(103 � 10�5) Associative Property

� 13.4 � 10�2 Product of Powers

� (1.34 � 101) � 10�2 13.4 � 1.34 � 101

� 1.34 � (101 � 10�2) Associative Property

� 1.34 � 10�1 or 0.134 Product of Powers

The solution is 1.34 � 10�1 or 0.134.

Evaluate . Express the result in scientific and

standard notation.

� � �� Associative Property

� 0.368 � 103 Quotient of Powers

� (3.68 � 10�1) � 103 0.368 � 3.68 � 10�1

� 3.68 � (10�1 � 103) Associative Property

� 3.68 � 102 or 368 Product of Powers

The solution is 3.68 � 102 or 368.

Evaluate. Express each result in scientific and standard notation.

1. 2. 3. (3.2 � 10�2)(2.0 � 102)

4. 5. (7.7 � 105)(2.1 � 102) 6.

7. (3.3 � 105)(1.5 � 10�4) 8. 9.

10. FUEL CONSUMPTION North America burned 4.5 � 1016 BTU of petroleum in 1998.At this rate, how many BTU’s will be burned in 9 years? Source: The New York Times 2001 Almanac

11. OIL PRODUCTION If the United States produced 6.25 � 109 barrels of crude oil in1998, and Canada produced 1.98 � 109 barrels, what is the quotient of their productionrates? Write a statement using this quotient. Source: The New York Times 2001 Almanac

4 � 10�4��2.5 � 102

3.3 � 10�12��1.1 � 10�14

9.72 � 108��7.2 � 1010

1.2672 � 10�8��

2.4 � 10�12

3 � 10�12��2 � 10�15

1.4 � 104��2 � 102

108�105

1.5088�4.1

1.5088 � 108��4.1 � 105

1.5088 � 108��

4.1 � 105


Scientific Notation

NAME ______________________________________________ DATE ____________ PERIOD _____

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Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeScientific Notation

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1. 4 � 103 2. 2 � 108 3. 3.2 � 105

4. 3 � 10�6 5. 9 � 10�2 6. 4.7 � 10�7

ASTRONOMY Express the number in each statement in standard notation.

7. The diameter of Jupiter is 1.42984 � 105 kilometers.

8. The surface density of the main ring around Jupiter is 5 � 10�6 grams per centimetersquared.

9. The minimum distance from Mars to Earth is 5.45 � 107 kilometers.


10. 41,000,000 11. 65,100 12. 283,000,000

13. 264,701 14. 0.019 15. 0.000007

16. 0.000010035 17. 264.9 18. 150 � 102


19. (3.1 � 107)(2 � 10�5) 20. (5 � 10�2)(1.4 � 10�4)

21. (3 � 103)(4.2 � 10�1) 22. (3 � 10�2)(5.2 � 109)

23. (2.4 � 102)(4 � 10�10) 24. (1.5 � 10�4)(7 � 10�5)

25. 26.7.2 � 10�5��4 � 10�3

5.1 � 106��1.5 � 102



1. 7.3 � 107 2. 2.9 � 103 3. 9.821 � 1012

4. 3.54 � 10�1 5. 7.3642 � 104 6. 4.268 � 10�6

PHYSICS Express the number in each statement in standard notation.

7. An electron has a negative charge of 1.6 � 10�19 Coulomb.

8. In the middle layer of the sun’s atmosphere, called the chromosphere, the temperatureaverages 2.78 � 104 degrees Celsius.


9. 915,600,000,000 10. 6387 11. 845,320 12. 0.00000000814

13. 0.00009621 14. 0.003157 15. 30,620 16. 0.0000000000112

17. 56 � 107 18. 4740 � 105 19. 0.076 � 10�3 20. 0.0057 � 103


21. (5 � 10�2)(2.3 � 1012) 22. (2.5 � 10�3)(6 � 1015)

23. (3.9 � 103)(4.2 � 10�11) 24. (4.6 � 10�4)(3.1 � 10�1)

25. 26. 27.

28. 29. 30.

31. BIOLOGY A cubic millimeter of human blood contains about 5 � 106 red blood cells. Anadult human body may contain about 5 � 106 cubic millimeters of blood. About how manyred blood cells does such a human body contain?

32. POPULATION The population of Arizona is about 4.778 � 106 people. The land area isabout 1.14 � 105 square miles. What is the population density per square mile?

2.015 � 10�3��

3.1 � 102

1.68 � 104��8.4 � 10�4

1.82 � 105��9.1 � 107

1.17 � 102��5 � 10�1

6.72 � 103��4.2 � 108

3.12 � 103��1.56 � 10�3

Practice Scientific Notation

NAME ______________________________________________ DATE ____________ PERIOD _____

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Reading to Learn MathematicsScientific Notation

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Pre-Activity Why is scientific notation important in astronomy?


In the table, each mass is written as the product of a number and a powerof 10. Look at the first factor in each product. How are these factors alike?

Reading the Lesson

1. Is the number 0.0543 � 104 in scientific notation? Explain.

2. Complete each sentence to change from scientific notation to standard notation.

a. To express 3.64 � 106 in standard notation, move the decimal point

places to the .

b. To express 7.825 � 10�3 in standard notation, move the decimal point

places to the .

3. Complete each sentence to change from standard notation to scientific notation.

a. To express 0.0007865 in scientific notation, move the decimal point places

to the right and write .

b. To express 54,000,000,000 in scientific notation, move the decimal point

places to the left and write .

4. Write positive or negative to complete each sentence.

a. powers of 10 are used to express very large numbers in

scientific notation.

b. powers of 10 are used to express very small numbers in

scientific notation.


5. Describe the method you would use to estimate how many times greater the mass ofSaturn is than the mass of Pluto.


Converting Metric UnitsScientific notation is convenient to use for unit conversions in the metric system.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3

How many kilometersare there in 4,300,000 meters?

Divide the measure by the number ofmeters (1000) in one kilometer. Expressboth numbers in scientific notation.

� 4.3 � 103

The answer is 4.3 3 103 km.

4.3 � 106��1 � 103

Convert 3700 grams intomilligrams.

Multiply by the number of milligrams(1000) in 1 gram.

(3.7 � 103)(1 � 103) � 3.7 � 106

There are 3.7 � 106 mg in 3700 g.


Complete the following. Express each answer in scientific notation.

1. 250,000 m � km 2. 375 km � m

3. 247 m � cm 4. 5000 m � mm

5. 0.0004 km � m 6. 0.01 mm � m

7. 6000 m � mm 8. 340 cm � km

9. 52,000 mg � g 10. 420 kL � L

Solve.

11. The planet Mars has a diameter of 6.76 � 103 km. What is thediameter of Mars in meters? Express the answer in both scientificand decimal notation.

12. The distance from earth to the sun is 149,590,000 km. Lighttravels 3.0 � 108 meters per second. How long does it take lightfrom the sun to reach the earth in minutes? Round to the nearest hundredth.

13. A light-year is the distance that light travels in one year. (SeeExercise 12.) How far is a light year in kilometers? Express youranswer in scientific notation. Round to the nearest hundredth.

Study Guide and InterventionPolynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

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Degree of a Polynomial A polynomial is a monomial or a sum of monomials. Abinomial is the sum of two monomials, and a trinomial is the sum of three monomials.Polynomials with more than three terms have no special name. The degree of a monomialis the sum of the exponents of all its variables. The degree of the polynomial is the sameas the degree of the monomial term with the highest degree.

State whether each expression is a polynomial. If the expression isa polynomial, identify it as a monomial, binomial, or trinomial. Then give thedegree of the polynomial.

Expression Polynomial?Monomial, Binomial, Degree of the

or Trinomial? Polynomial

3x � 7xyzYes. 3x � 7xyz � 3x (�7xyz),

binomial 3which is the sum of two monomials

�25 Yes. �25 is a real number. monomial 0

7n3 3n�4No. 3n�4 � , which is not

none of these —a monomial

Yes. The expression simplifies to9x3 4x x 4 2x 9x3 7x 4, which is the sum trinomial 3

of three monomials

State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, binomial, or trinomial.

1. 36 2. 5

3. 7x � x 5 4. 8g2h � 7gh 2

5. 5y � 8 6. 6x x2

Find the degree of each polynomial.

7. 4x2y3z 8. �2abc 9. 15m

10. s 5t 11. 22 12. 18x2 4yz � 10y

13. x4 � 6x2 � 2x3 � 10 14. 2x3y2 � 4xy3 15. �2r8s4 7r2s � 4r7s6

16. 9x2 yz8 17. 8b bc5 18. 4x4y � 8zx2 2x5

19. 4x2 � 1 20. 9abc bc � d5 21. h3m 6h4m2 � 7

1�4y2

3�q2

3�n4

ExampleExample

ExercisesExercises


Write Polynomials in Order The terms of a polynomial are usually arranged so that the powers of one variable are in ascending (increasing) order or descending(decreasing) order.


Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4

Arrange the terms ofeach polynomial so that the powers ofx are in ascending order.

a. x4 � x2 � 5x3

�x2 5x3 x4

b. 8x3y � y2 � 6x2y � xy2

�y2 xy2 6x2y 8x3y

Arrange the terms ofeach polynomial so that the powers ofx are in descending order.

a. x4 � 4x5 � x2

4x5 x4 � x2

b. �6xy � y3 � x2y2 � x4y2

x4y2 � x2y2 � 6xy y3


ExercisesExercises

Arrange the terms of each polynomial so that the powers of x are in ascending order.

1. 5x x2 6 2. 6x 9 � 4x2 3. 4xy 2y 6x2

4. 6y2x � 6x2y 2 5. x4 x3 x2 6. 2x3 � x 3x7

7. �5cx 10c2x3 15cx2 8. �4nx � 5n3x3 5 9. 4xy 2y 5x2

Arrange the terms of each polynomial so that the powers of x are in descending order.

10. 2x x2 � 5 11. 20x � 10x2 5x3 12. x2 4yx� 10x5

13. 9bx 3bx2 � 6x3 14. x3 x5 � x2 15. ax2 8a2x5 � 4

16. 3x3y � 4xy2 � x4y2 y5 17. x4 4x3 � 7x5 1

18. �3x6 � x5 2x8 19. �15cx2 8c2x5 cx

20. 24x2y � 12x3y2 6x4 21. �15x3 10x4y2 7xy2

Skills PracticePolynomials

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State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, a binomial, or a trinomial.

1. 5mn n2 2. 4by 2b � by 3. �32

4. 5. 5x2 � 3x�4 6. 2c2 8c 9 � 3

GEOMETRY Write a polynomial to represent the area of each shaded region.

7. 8.


9. 12 10. 3r4 11. b 6

12. 4a3 � 2a 13. 5abc � 2b2 1 14. 8x5y4 � 2x8

Arrange the terms of each polynomial so that the powers of x are in ascendingorder.

15. 3x 1 2x2 16. 5x � 6 3x2

17. 9x2 2 x3 x 18. �3 3x3 � x2 4x

19. 7r5x 21r4 � r2x2 � 15x3 20. 3a2x4 14a2 � 10x3 ax2

Arrange the terms of each polynomial so that the powers of x are in descendingorder.

21. x2 3x3 27 � x 22. 25 � x3 x

23. x � 3x2 4 5x3 24. x2 64 � x 7x3

25. 2cx 32 � c3x2 6x3 26. 13 � x3y3 x2y2 x

r

a

x

by

3x�7


State whether each expression is a polynomial. If the expression is a polynomial,identify it as a monomial, a binomial, or a trinomial.

1. 7a2b 3b2 � a2b 2. y3 y2 � 9 3. 6g2h3k

GEOMETRY Write a polynomial to represent the area of each shaded region.

4. 5.


6. x 3x4 � 21x2 x3 7. 3g2h3 g3h

8. �2x2y 3xy3 x2 9. 5n3m � 2m3 n2m4 n2

10. a3b2c 2a5c b3c2 11. 10s2t2 4st2 � 5s3t2

Arrange the terms of each polynomial so that the powers of x are in ascendingorder.

12. 8x2 � 15 5x5 13. 10bx � 7b2 x4 4b2x3

14. �3x3y 8y2 xy4 15. 7ax � 12 3ax3 a2x2

Arrange the terms of each polynomial so that the powers of x are in descendingorder.

16. 13x2 � 5 6x3 � x 17. 4x 2x5 � 6x3 2

18. g2x � 3gx3 7g3 4x2 19. �11x2y3 6y � 2xy 2x4

20. 7a2x2 17 � a3x3 2ax 21. 12rx3 9r6 r2x 8x6

22. MONEY Write a polynomial to represent the value of t ten-dollar bills, f fifty-dollarbills, and h one-hundred-dollar bills.

23. GRAVITY The height above the ground of a ball thrown up with a velocity of 96 feet persecond from a height of 6 feet is 6 96t � 16t2 feet, where t is the time in seconds.According to this model, how high is the ball after 7 seconds? Explain.

da

b

b

1�5

Practice Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4

Reading to Learn MathematicsPolynomials

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Pre-Activity How are polynomials useful in modeling data?


• How many terms does t4 � 9t3 � 26t � 18t � 76 have?

• What could you call a polynomial with just one term?

Reading the Lesson

1. What is the meaning of the prefixes mono-, bi-, and tri-?

2. Write examples of words that begin with the prefixes mono-, bi-, and tri-.

3. Complete the table.

monomial binomial trinomialpolynomial with more

than three terms

Example 3r 2t 2x2 � 3x 5x2 � 3x � 2 7s2 � s4 � 2s3 � s � 5

Number of Terms

4. What is the degree of the monomial 3xy2z?

5. What is the degree of the polynomial 4x4 � 2x3y3 � y2 � 14? Explain how you foundyour answer.


6. Use a dictionary to find the meaning of the terms ascending and descending. Write theirmeanings and then describe a situation in your everyday life that relates to them.


Polynomial FunctionsSuppose a linear equation such as 23x � y � 4 is solved for y. Then an equivalent equation,y � 3x � 4, is found. Expressed in this way, y is a function of x, or f(x) � 3x � 4. Notice that the right side of the equation is a binomial of degree 1.

Higher-degree polynomials in x may also form functions. An example is f(x) � x3 � 1, which is a polynomial function of degree 3. You can graph this function using a table of ordered pairs, as shown at the right.

For each of the following polynomial functions, make a tableof values for x and y � f(x). Then draw the graph on the grid.

1. f(x) � 1 � x2 2. f(x) � x2 � 5

3. f(x) � x2 � 4x � 1 4. f(x) � x3

x

y

Ox

y

O

x

y

Ox

y

O

x

y

O

x y

�1�12

� �2�38

�

�1 0

0 1

1 2

1�12

� 4 �38

�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4

Study Guide and InterventionAdding and Subtracting Polynomials

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6-58-5


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Add Polynomials To add polynomials, you can group like terms horizontally or writethem in column form, aligning like terms vertically. Like terms are monomial terms thatare either identical or differ only in their coefficients, such as 3p and �5p or 2x2y and 8x2y.

Find (2x2 � x � 8) �(3x � 4x2 � 2).

Horizontal MethodGroup like terms.

(2x2 � x � 8) � (3x � 4x2 � 2)� [(2x2 � (�4x2)] � (x � 3x ) � [(�8) � 2)]� �2x2 � 4x � 6.

The sum is �2x2 � 4x � 6.

Find (3x2 � 5xy) �(xy � 2x2).

Vertical MethodAlign like terms in columns and add.

3x2 � 5xy(�) 2x2 � xy Put the terms in descending order.

5x2 � 6xy

The sum is 5x2 � 6xy.


ExercisesExercises

Find each sum.

1. (4a � 5) � (3a � 6) 2. (6x � 9) � (4x2 � 7)

3. (6xy � 2y � 6x) � (4xy � x) 4. (x2 � y2) � (�x2 � y2)

5. (3p2 � 2p � 3) � (p2 � 7p � 7) 6. (2x2 � 5xy � 4y2) � (�xy � 6x2 � 2y2)

7. (5p � 2q) � (2p2 � 8q � 1) 8. (4x2 � x � 4) � (5x � 2x2 � 2)

9. (6x2 � 3x) � (x2 � 4x � 3) 10. (x2 � 2xy � y2) � (x2 � xy � 2y2)

11. (2a � 4b � c) � (�2a � b � 4c) 12. (6xy2 � 4xy) � (2xy � 10xy2 � y2)

13. (2p � 5q) � (3p � 6q) � (p � q) 14. (2x2 � 6) � (5x2 � 2) � (�x2 � 7)

15. (3z2 � 5z) � (z2 � 2z) � (z � 4) 16. (8x2 � 4x � 3y2 � y) � (6x2 � x � 4y)


Subtract Polynomials You can subtract a polynomial by adding its additive inverse.To find the additive inverse of a polynomial, replace each term with its additive inverse or opposite.

Find (3x2 � 2x � 6) � (2x � x2 � 3).


Adding and Subtracting Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

ExampleExample

Horizontal Method

Use additive inverses to rewrite as addition.Then group like terms.

(3x2 � 2x � 6) � (2x � x2 � 3)� (3x2 � 2x � 6) � [(�2x)� (�x2) � (�3)]� [3x2 � (�x2)] � [2x � (�2x)] � [�6 � (�3)]� 2x2 � (�9)� 2x2 � 9

The difference is 2x2 � 9.

Vertical Method

Align like terms in columns andsubtract by adding the additive inverse.

3x2 � 2x � 6(�) x2 � 2x � 3

3x2 � 2x � 6(�) �x2 � 2x � 3

2x2 � 9The difference is 2x2 � 9.

ExercisesExercises

Find each difference.

1. (3a � 5) � (5a � 1) 2. (9x � 2) � (�3x2 � 5)

3. (9xy � y � 2x) � (6xy � 2x) 4. (x2 � y2) � (�x2 � y2)

5. (6p2 � 4p � 5) � (2p2 � 5p � 1) 6. (6x2 � 5xy � 2y2) � (�xy � 2x2 � 4y2)

7. (8p � 5q) � (�6p2 � 6q � 3) 8. (8x2 � 4x � 3) � (�2x � x2 � 5)

9. (3x2 � 2x) � (3x2 � 5x � 1) 10. (4x2 � 6xy � 2y2) � (�x2 � 2xy � 5y2)

11. (2h � 6j � 2k) � (�7h � 5j � 4k) 12. (9xy2 � 5xy) � (�2xy � 8xy2)

13. (2a � 8b) � (�3a � 5b) 14. (2x2 � 8) � (�2x2 � 6)

15. (6z2 � 4z � 2) � (4z2 � z) 16. (6x2 � 5x � 1) � (�7x2 � 2x � 4)

Skills PracticeAdding and Subtracting Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5


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Find each sum or difference.

1. (2x � 3y) � (4x � 9y) 2. (6s � 5t) � (4t � 8s)

3. (5a � 9b) � (2a � 4b) 4. (11m � 7n) � (2m � 6n)

5. (m2 � m) � (2m � m2) 6. (x2 � 3x) � (2x2 � 5x)

7. (d2 � d � 5) � (2d � 5) 8. (2e2 � 5e) � (7e � 3e2)

9. (5f � g � 2) � (�2f � 3) 10. (6k2 � 2k � 9) � (4k2 � 5k)

11. (x3 � x � 1) � (3x � 1) 12. (b2 � ab � 2) � (2b2 � 2ab)

13. (7z2 � 4 � z) � (�5 � 3z2) 14. (5 � 4n � 2m) � (�6m � 8)

15. (4t2 � 2) � (�4 � 2t) 16. (3g3 � 7g) � (4g � 8g3)

17. (2a2 � 8a � 4) � (a2 � 3) 18. (3x2 � 7x � 5) � (�x2 � 4x)

19. (7z2 � z � 1) � (�4z � 3z2 � 3) 20. (2c2 � 7c � 4) � (c2 � 1 � 9c)

21. (n2 � 3n � 2) � (2n2 � 6n � 2) 22. (a2 � ab � 3b2) � (b2 � 4a2 � ab)

23. (�2 � 5� � 6) � (2�2 � 5 � �) 24. (2m2 � 5m � 1) � (4m2 � 3m � 3)

25. (x2 � 6x � 2) � (�5x2 � 7x � 4) 26. (5b2 � 9b � 5) � (b2 � 6 � 2b)

27. (2x2 � 6x � 2) � (x2 � 4x) � (3x2 � x � 5)



1. (4y � 5) � (�7y � 1) 2. (�x2 � 3x) � (5x � 2x2)

3. (4k2 � 8k � 2) � (2k � 3) 4. (2m2 � 6m) � (m2 � 5m � 7)

5. (2w2 � 3w � 1) � (4w � 7) 6. (g3 � 2g2) � (6g � 4g2 � 2g3)

7. (5a2 � 6a � 2) � (7a2 � 7a � 5) 8. (�4p2 � p � 9) � (p2 � 3p � 1)

9. (x3 � 3x � 1) � (x3 � 7 � 12x) 10. (6c2 � c � 1) � (�4 � 2c2 � 8c)

11. (�b3 � 8bc2 � 5) � (7bc2 � 2 � b3) 12. (5n2 � 3n � 2) � (�n � 2n2 � 4)

13. (4y2 � 2y � 8) � (7y2 � 4 � y) 14. (w2 � 4w � 1) � (�5 � 5w2 � 3w)

15. (4u2 � 2u � 3) � (3u2 � u � 4) 16. (5b2 � 8 � 2b) � (b � 9b2 � 5)

17. (4d2 � 2d � 2) � (5d2 � 2 � d) 18. (8x2 � x � 6) � (�x2 � 2x � 3)

19. (3h2 � 7h � 1) � (4h � 8h2 � 1) 20. (4m2 � 3m � 10) � (m2 � m � 2)

21. (x2 � y2 � 6) � (5x2 � y2 � 5) 22. (7t2 � 2 � t) � (t2 � 7 � 2t)

23. (k3 � 2k2 � 4k � 6) � (�4k � k2 � 3) 24. (9j2 � j � jk) � (�3j2 � jk � 4j)

25. (2x � 6y � 3z) � (4x � 6z � 8y) � (x � 3y � z)

26. (6f 2 � 7f � 3) � (5f 2 � 1 � 2f ) � (2f 2 � 3 � f )

27. BUSINESS The polynomial s3 � 70s2 � 1500s � 10,800 models the profit a companymakes on selling an item at a price s. A second item sold at the same price brings in aprofit of s3 � 30s2 � 450s � 5000. Write a polynomial that expresses the total profitfrom the sale of both items.

28. GEOMETRY The measures of two sides of a triangle are given.If P is the perimeter, and P � 10x � 5y, find the measure of the third side.

3x � 4y

5x � y

Practice Adding and Subtracting Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

Reading to Learn MathematicsAdding and Subtracting Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5


Less

on

8-5

Pre-Activity How can adding polynomials help you model sales?


What operation would you use to find how much more the traditional toysales R were than the video games sales V ?

Reading the Lesson

1. Use the example (�3x3 � 4x2 � 5x � 1) � (�5x3 � 2x2 � 2x � 7).

a. Show what is meant by grouping like terms horizontally.

b. Show what is meant by aligning like terms vertically.

c. Choose one method, then add the polynomials.

2. How is subtracting a polynomial like subtracting a rational number?

3. An algebra student got the following exercise wrong on his homework. What was his error?

(3x5 � 3x4 � 2x3 � 4x2 � 5) � (2x5 � x3 � 2x2 � 4)� [3x5 � (�2x5)] � (�3x4) � [2x3 � (�x3)] � [�4x2 � (�2x2)] � (5 � 4)� x5 � 3x4 � x3 � 6x2 � 9


4. How is adding and subtracting polynomials vertically like adding and subtractingdecimals vertically?


Circular Areas and Volumes

Area of Circle Volume of Cylinder Volume of Cone

A � �r2 V � �r2h V � �r2h

Write an algebraic expression for each shaded area. (Recall that the diameter of a circle is twice its radius.)

1. 2. 3.

Write an algebraic expression for the total volume of each figure.

4. 5.

Each figure has a cylindrical hole with a radius of 2 inches and a height of 5 inches. Find each volume.

6. 7.

3x

4x

5x

7x

3x—2

x � a

x

x � b

x

2x5x

3x2x2x x

y x x yx

x

r

h

r

h

r

1�3

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

Study Guide and InterventionMultiplying a Polynomial by a Monomial

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6


Less

on

8-6

Product of Monomial and Polynomial The Distributive Property can be used tomultiply a polynomial by a monomial. You can multiply horizontally or vertically. Sometimesmultiplying results in like terms. The products can be simplified by combining like terms.

Find �3x2(4x2 � 6x � 8).

Horizontal Method�3x2(4x2 � 6x � 8)

� �3x2(4x2) � (�3x2)(6x) � (�3x2)(8)� �12x4 � (�18x3) � (�24x2)� �12x4 � 18x3 � 24x2

Vertical Method4x2 � 6x � 8

(�) �3x2

�12x4 � 18x3 � 24x2

The product is �12x4 � 18x3 � 24x2.

Simplify �2(4x2 � 5x) �x(x2 � 6x).

�2(4x2 � 5x) � x( x2 � 6x)� �2(4x2) � (�2)(5x) � (�x)(x2) � (�x)(6x)� �8x2 � (�10x) � (�x3) � (�6x2)� (�x3) � [�8x2 � (�6x2)] � (�10x)� �x3 � 14x2 � 10x


ExercisesExercises

Find each product.

1. x(5x � x2) 2. x(4x2 � 3x � 2) 3. �2xy(2y � 4x2)

4. �2g( g2 � 2g � 2) 5. 3x(x4 � x3� x2) 6. �4x(2x3 � 2x � 3)

7. �4cx(10 � 3x) 8. 3y(�4x � 6x3� 2y) 9. 2x2y2(3xy � 2y � 5x)

Simplify.

10. x(3x � 4) � 5x 11. �x(2x2 � 4x) � 6x2

12. 6a(2a � b) � 2a(�4a � 5b) 13. 4r(2r2 � 3r � 5) � 6r(4r2 � 2r � 8)

14. 4n(3n2 � n � 4) � n(3 � n) 15. 2b(b2 � 4b � 8) � 3b(3b2 � 9b � 18)

16. �2z(4z2 � 3z � 1) � z(3z2 � 2z � 1) 17. 2(4x2 � 2x) � 3(�6x2 � 4) � 2x(x � 1)


Solve Equations with Polynomial Expressions Many equations containpolynomials that must be added, subtracted, or multiplied before the equation can be solved.

Solve 4(n � 2) � 5n � 6(3 � n) � 19.

4(n � 2) � 5n � 6(3 � n) � 19 Original equation

4n � 8 � 5n � 18 � 6n � 19 Distributive Property

9n � 8 � 37 � 6n Combine like terms.

15n � 8 � 37 Add 6n to both sides.

15n � 45 Add 8 to both sides.

n � 3 Divide each side by 15.

The solution is 3.

Solve each equation.

1. 2(a � 3) � 3(�2a � 6) 2. 3(x � 5) � 6 � 18

3. 3x(x � 5) � 3x2 � �30 4. 6(x2 � 2x) � 2(3x2 � 12)

5. 4(2p � 1) � 12p � 2(8p � 12) 6. 2(6x � 4) � 2 � 4(x � 4)

7. �2(4y � 3) � 8y � 6 � 4(y � 2) 8. c(c � 2) � c(c � 6) � 10c � 12

9. 3(x2 � 2x) � 3x2 � 5x � 11 10. 2(4x � 3) � 2 � �4(x � 1)

11. 3(2h � 6) � (2h � 1) � 9 12. 3(y � 5) � (4y � 8) � �2y � 10

13. 3(2a � 6) � (�3a � 1) � 4a � 2 14. 5(2x2 � 1) � (10x2 � 6) � �(x � 2)

15. 3(x � 2) � 2(x � 1) � �5(x � 3) 16. 4(3p2 � 2p) � 12p2 � 2(8p � 6)


Multiplying a Polynomial by a Monomial

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8-68-6

ExampleExample

ExercisesExercises

Skills PracticeMultiplying a Polynomial by a Monomial

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8-6

Find each product.

1. a(4a � 3) 2. �c(11c � 4)

3. x(2x � 5) 4. 2y(y � 4)

5. �3n(n2 � 2n) 6. 4h(3h � 5)

7. 3x(5x2 � x � 4) 8. 7c(5 � 2c2 � c3)

9. �4b(1 � 9b � 2b2) 10. 6y(�5 � y � 4y2)

11. 2m2(2m2 � 3m � 5) 12. �3n2(�2n2 � 3n � 4)

Simplify.

13. w(3w � 2) � 5w 14. f(5f � 3) � 2f

15. �p(2p � 8) � 5p 16. y2(�4y � 5) � 6y2

17. 2x(3x2 � 4) � 3x3 18. 4a(5a2 � 4) � 9a

19. 4b(�5b � 3) � 2(b2 � 7b � 4) 20. 3m(3m � 6) � 3(m2 � 4m � 1)


21. 3(a � 2) � 5 � 2a � 4 22. 2(4x � 2) � 8 � 4(x � 3)

23. 5( y � 1) � 2 � 4( y � 2) � 6 24. 4(b � 6) � 2(b � 5) � 2

25. 6(m � 2) � 14 � 3(m � 2) � 10 26. 3(c � 5) � 2 � 2(c � 6) � 2


Find each product.

1. 2h(�7h2 � 4h) 2. 6pq(3p2 � 4q) 3. �2u2n(4u � 2n)

4. 5jk(3jk � 2k) 5. �3rs(�2s2 � 3r) 6. 4mg2(2mg � 4g)

7. � m(8m2 � m � 7) 8. � n2(�9n2 � 3n � 6)

Simplify.

9. �2�(3� � 4) � 7� 10. 5w(�7w � 3) � 2w(�2w2 � 19w � 2)

11. 6t(2t � 3) � 5(2t2 � 9t � 3) 12. �2(3m3 � 5m � 6) � 3m(2m2 � 3m � 1)

13. �3g(7g � 2) � 3(g2 � 2g � 1) � 3g(�5g � 3)

14. 4z2(z � 7) � 5z(z2 � 2z � 2) � 3z(4z � 2)


15. 5(2s � 1) � 3 � 3(3s � 2) 16. 3(3u � 2) � 5 � 2(2u � 2)

17. 4(8n � 3) � 5 � 2(6n � 8) � 1 18. 8(3b � 1) � 4(b � 3) � 9

19. h(h � 3) � 2h � h(h � 2) � 12 20. w(w � 6) � 4w � �7 � w(w � 9)

21. t(t � 4) � 1 � t(t � 2) � 2 22. u(u � 5) � 8u � u(u � 2) � 4

23. NUMBER THEORY Let x be an integer. What is the product of twice the integer addedto three times the next consecutive integer?

INVESTMENTS For Exercises 24–26, use the following information.Kent invested $5,000 in a retirement plan. He allocated x dollars of the money to a bondaccount that earns 4% interest per year and the rest to a traditional account that earns 5%interest per year.

24. Write an expression that represents the amount of money invested in the traditionalaccount.

25. Write a polynomial model in simplest form for the total amount of money T Kent hasinvested after one year. (Hint: Each account has A � IA dollars, where A is the originalamount in the account and I is its interest rate.)

26. If Kent put $500 in the bond account, how much money does he have in his retirementplan after one year?

2�3

1�4

Practice Multiplying a Polynomial by a Monomial

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6

Reading to Learn MathematicsMultiplying a Polynomial by a Monomial

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6


Less

on

8-6

Pre-Activity How is finding the product of a monomial and a polynomial relatedto finding the area of a rectangle?


You may recall that the formula for the area of a rectangle is A � �w. In

this rectangle, � � and w � . How would you

substitute these values in the area formula?

Reading the Lesson

1. Refer to Lesson 8-6.

a. How is the Distributive Property used to multiply a polynomial by a monomial?

b. Use the Distributive Property to complete the following.

2y2(3y2 � 2y � 7) � 2y2( ) � 2y2( ) � 2y2( )

� � �

�3x3(x3 � 2x2 � 3) � � �

� � �

2. What is the difference between simplifying an expression and solving an equation?


3. Use the equation 2x(x � 5) � 3x(x � 3) � 5x(x � 7) � 9 to show how you would explainthe process of solving equations with polynomial expressions to another algebra student.


Figurate NumbersThe numbers below are called pentagonal numbers. They are the numbers of dots or disksthat can be arranged as pentagons.

1. Find the product n(3n � 1).

2. Evaluate the product in Exercise 1 for values of n from 1 through 4.

3. What do you notice?

4. Find the next six pentagonal numbers.

5. Find the product n(n � 1).

6. Evaluate the product in Exercise 5 for values of n from 1 through 5. On another sheet ofpaper, make drawings to show why these numbers are called the triangular numbers.

7. Find the product n(2n � 1).

8. Evaluate the product in Exercise 7 for values of n from 1 through 5. Draw thesehexagonal numbers.

9. Find the first 5 square numbers. Also, write the general expression for any squarenumber.

The numbers you have explored above are called the plane figurate numbers because theycan be arranged to make geometric figures. You can also create solid figurate numbers.

10. If you pile 10 oranges into a pyramid with a triangle as a base, you get one of thetetrahedral numbers. How many layers are there in the pyramid? How many orangesare there in the bottom layers?

11. Evaluate the expression n3 � n2 � n for values of n from 1 through 5 to find the

first five tetrahedral numbers.

1�3

1�2

1�6

1�2

1�2

●●

221251

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6

Study Guide and InterventionMultiplying Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7


Less

on

8-7

Multiply Binomials To multiply two binomials, you can apply the Distributive Propertytwice. A useful way to keep track of terms in the product is to use the FOIL method asillustrated in Example 2.

Find (x � 3)(x � 4).

Horizontal Method(x � 3)(x � 4)

� x(x � 4) � 3(x � 4)� (x)(x) � x(�4) � 3(x)� 3(�4)� x2 � 4x � 3x � 12� x2 � x � 12

Vertical Method

x � 3(�) x � 4

�4x � 12x2 � 3xx2 � x � 12

The product is x2 � x � 12.

Find (x � 2)(x � 5) usingthe FOIL method.

(x � 2)(x � 5)First Outer Inner Last

� (x)(x) � (x)(5) � (�2)(x) � (�2)(5)� x2 � 5x � (�2x) � 10� x2 � 3x � 10

The product is x2 � 3x � 10.


ExercisesExercises

Find each product.

1. (x � 2)(x � 3) 2. (x � 4)(x � 1) 3. (x � 6)(x � 2)

4. (p � 4)(p � 2) 5. (y � 5)(y � 2) 6. (2x � 1)(x � 5)

7. (3n � 4)(3n � 4) 8. (8m � 2)(8m � 2) 9. (k � 4)(5k � 1)

10. (3x � 1)(4x � 3) 11. (x � 8)(�3x � 1) 12. (5t � 4)(2t � 6)

13. (5m � 3n)(4m � 2n) 14. (a � 3b)(2a � 5b) 15. (8x � 5)(8x � 5)

16. (2n � 4)(2n � 5) 17. (4m � 3)(5m � 5) 18. (7g � 4)(7g � 4)


Multiply Polynomials The Distributive Property can be used to multiply any two polynomials.

Find (3x � 2)(2x2 � 4x � 5).

(3x � 2)(2x2 � 4x � 5)� 3x(2x2 � 4x � 5) � 2(2x2 � 4x � 5) Distributive Property

� 6x3 � 12x2 � 15x � 4x2 � 8x � 10 Distributive Property

� 6x3 � 8x2 � 7x � 10 Combine like terms.

The product is 6x3 � 8x2 � 7x � 10.

Find each product.

1. (x � 2)(x2 � 2x � 1) 2. (x � 3)(2x2 � x � 3)

3. (2x � 1)(x2 � x � 2) 4. (p � 3)(p2 � 4p � 2)

5. (3k � 2)(k2 � k � 4) 6. (2t � 1)(10t2 � 2t � 4)

7. (3n � 4)(n2 � 5n � 4) 8. (8x � 2)(3x2 � 2x � 1)

9. (2a � 4)(2a2 � 8a � 3) 10. (3x � 4)(2x2 � 3x � 3)

11. (n2 � 2n � 1)(n2 � n � 2) 12. (t2 � 4t � 1)(2t2 � t � 3)

13. (y2 � 5y � 3)(2y2 � 7y � 4) 14. (3b2 � 2b � 1)(2b2 � 3b � 4)


Multiplying Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

ExampleExample

ExercisesExercises

Skills PracticeMultiplying Polynomials

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8-78-7


Less

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8-7

Find each product.

1. (m � 4)(m � 1) 2. (x � 2)(x � 2)

3. (b � 3)(b � 4) 4. (t � 4)(t � 3)

5. (r � 1)(r � 2) 6. (z � 5)(z � 1)

7. (3c � 1)(c � 2) 8. (2x � 6)(x � 3)

9. (d � 1)(5d � 4) 10. (2� � 5)(� � 4)

11. (3n � 7)(n � 3) 12. (q � 5)(5q � 1)

13. (3b � 3)(3b � 2) 14. (2m � 2)(3m � 3)

15. (4c � 1)(2c � 1) 16. (5a � 2)(2a � 3)

17. (4h � 2)(4h � 1) 18. (x � y)(2x � y)

19. (e � 4)(e2 � 3e � 6) 20. (t � 1)(t2 � 2t � 4)

21. (k � 4)(k2 � 3k � 6) 22. (m � 3)(m2 � 3m � 5)

GEOMETRY Write an expression to represent the area of each figure.

23. 24.

2x � 4

2x � 5

4x � 1


Find each product.

1. (q � 6)(q � 5) 2. (x � 7)(x � 4) 3. (s � 5)(s � 6)

4. (n � 4)(n � 6) 5. (a � 5)(a � 8) 6. (w � 6)(w � 9)

7. (4c � 6)(c � 4) 8. (2x � 9)(2x � 4) 9. (4d � 5)(2d � 3)

10. (4b � 3)(3b � 4) 11. (4m � 2)(4m � 3) 12. (5c � 5)(7c � 9)

13. (6a � 3)(7a � 4) 14. (6h � 3)(4h � 2) 15. (2x � 2)(5x � 4)

16. (3a � b)(2a � b) 17. (4g � 3h)(2g � 3h) 18. (4x � y)(4x � y)

19. (m � 5)(m2 � 4m � 8) 20. (t � 3)(t2 � 4t � 7)

21. (2h � 3)(2h2 � 3h � 4) 22. (3d � 3)(2d2 � 5d � 2)

23. (3q � 2)(9q2 � 12q � 4) 24. (3r � 2)(9r2 � 6r � 4)

25. (3c2 � 2c � 1)(2c2 � c � 9) 26. (2�2 � � � 3)(4�2 � 2� � 2)

27. (2x2 � 2x � 3)(2x2 � 4x � 3) 28. (3y2 � 2y � 2)(3y2 � 4y � 5)

GEOMETRY Write an expression to represent the area of each figure.

29. 30.

31. NUMBER THEORY Let x be an even integer. What is the product of the next twoconsecutive even integers?

32. GEOMETRY The volume of a rectangular pyramid is one third the product of the areaof its base and its height. Find an expression for the volume of a rectangular pyramidwhose base has an area of 3x2 � 12x � 9 square feet and whose height is x � 3 feet.

3x � 2

5x � 4

x � 1

4x � 2

2x � 2

Practice Multiplying Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

Reading to Learn MathematicsMultiplying Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7


Less

on

8-7

Pre-Activity How is multiplying binomials similar to multiplying two-digitnumbers?


In your own words, explain how the distributive property is used twice tomultiply two-digit numbers.

Reading the Lesson

1. How is multiplying binomials similar to multiplying two-digit numbers?

2. Complete the table using the FOIL method.

Product of�

Product of�

Product of�

Product of First Terms Outer Terms Inner Terms Last Terms

(x � 5)(x � 3) � � � �

� � � �

� � �

(3y � 6)(y � 2) � � � �

� � � �

� �


3. Think of a method for remembering all the product combinations used in the FOILmethod for multiplying two binomials. Describe your method using words or a diagram.


Pascal’s TriangleThis arrangement of numbers is called Pascal’s Triangle.It was first published in 1665, but was known hundreds of years earlier.

1. Each number in the triangle is found by adding two numbers.What two numbers were added to get the 6 in the 5th row?

2. Describe how to create the 6th row of Pascal’s Triangle.

3. Write the numbers for rows 6 through 10 of the triangle.

Row 6:

Row 7:

Row 8:

Row 9:

Row 10:

Multiply to find the expanded form of each product.

4. (a � b)2

5. (a � b)3

6. (a � b)4

Now compare the coefficients of the three products in Exercises 4–6 with Pascal’s Triangle.

7. Describe the relationship between the expanded form of (a � b)n and Pascal’s Triangle.

8. Use Pascal’s Triangle to write the expanded form of (a � b)6.

11 1

1 2 11 3 3 1

1 4 6 4 1

Enrichment

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Study Guide and InterventionSpecial Products

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8-88-8


Less

on

8-8

Squares of Sums and Differences Some pairs of binomials have products thatfollow specific patterns. One such pattern is called the square of a sum. Another is called thesquare of a difference.

Square of a sum (a � b)2 � (a � b)(a � b) � a2 � 2ab � b2

Square of a difference (a � b)2 � (a � b)(a � b) � a2 � 2ab � b2

Find (3a � 4)(3a � 4).

Use the square of a sum pattern, with a �3a and b � 4.

(3a � 4)(3a � 4) � (3a)2 � 2(3a)(4) � (4)2

� 9a2 � 24a � 16

The product is 9a2 � 24a � 16.

Find (2z � 9)(2z � 9).

Use the square of a difference pattern witha � 2z and b � 9.

(2z � 9)(2z � 9) � (2z)2 � 2(2z)(9) � (9)(9)� 4z2 � 36z � 81

The product is 4z2 � 36z � 81.


ExercisesExercises

Find each product.

1. (x � 6)2 2. (3p � 4)2 3. (4x � 5)2

4. (2x � 1)2 5. (2h � 3)2 6. (m � 5)2

7. (c � 3)2 8. (3 � p)2 9. (x � 5y)2

10. (8y � 4)2 11. (8 � x)2 12. (3a � 2b)2

13. (2x � 8)2 14. (x2 � 1)2 15. (m2 � 2)2

16. (x3 � 1)2 17. (2h2 � k2)2 18. � x � 3�2

19. (x � 4y2)2 20. (2p � 4q)2 21. � x � 2�22�3

1�4


Product of a Sum and a Difference There is also a pattern for the product of asum and a difference of the same two terms, (a � b)(a � b). The product is called thedifference of squares.

Product of a Sum and a Difference (a � b)(a � b) � a2 � b2

Find (5x � 3y)(5x � 3y).

(a � b)(a � b) � a2 � b2 Product of a Sum and a Difference

(5x � 3y)(5x � 3y) � (5x)2 � (3y)2 a � 5x and b � 3y

� 25x2 � 9y2 Simplify.

The product is 25x2 � 9y2.

Find each product.

1. (x � 4)(x � 4) 2. (p � 2)(p � 2) 3. (4x � 5)(4x � 5)

4. (2x � 1)(2x � 1) 5. (h � 7)(h � 7) 6. (m � 5)(m � 5)

7. (2c � 3)(2c � 3) 8. (3 � 5q)(3 � 5q) 9. (x � y)(x � y)

10. ( y � 4x)( y � 4x) 11. (8 � 4x)(8 � 4x) 12. (3a � 2b)(3a � 2b)

13. (3y � 8)(3y � 8) 14. (x2 � 1)(x2 � 1) 15. (m2 � 5)(m2 � 5)

16. (x3 � 2)(x3 � 2) 17. (h2 � k2)(h2 � k2) 18. � x � 2�� x � 2�

19. (3x � 2y2)(3x � 2y2) 20. (2p � 5s)(2p � 5s) 21. � x � 2y�� x � 2y�4�3

4�3

1�4

1�4


Special Products

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8-88-8

ExampleExample

ExercisesExercises

Skills PracticeSpecial Products

NAME ______________________________________________ DATE ____________ PERIOD _____

8-88-8


Less

on

8-8

Find each product.

1. (n � 3)2 2. (x � 4)(x � 4)

3. ( y � 7)2 4. (t � 3)(t � 3)

5. (b � 1)(b � 1) 6. (a � 5)(a � 5)

7. (p � 4)2 8. (z � 3)(z � 3)

9. (� � 2)(� � 2) 10. (r � 1)(r � 1)

11. (3g � 2)(3g � 2) 12. (2m � 3)(2m � 3)

13. (6 � u)2 14. (r � s)2

15. (3q � 1)(3q � 1) 16. (c � e)2

17. (2k � 2)2 18. (w � 3h)2

19. (3p � 4)(3p � 4) 20. (t � 2u)2

21. (x � 4y)2 22. (3b � 7)(3b � 7)

23. (3y � 3g)(3y � 3g) 24. (s2 � r2)2

25. (2k � m2)2 26. (3u2 � n)2

27. GEOMETRY The length of a rectangle is the sum of two whole numbers. The width ofthe rectangle is the difference of the same two whole numbers. Using these facts, write averbal expression for the area of the rectangle.


Find each product.

1. (n � 9)2 2. (q � 8)2 3. (� � 10)2

4. (r � 11)2 5. ( p � 7)2 6. (b � 6)(b � 6)

7. (z � 13)(z � 13) 8. (4e � 2)2 9. (5w � 4)2

10. (6h � 1)2 11. (3s � 4)2 12. (7v � 2)2

13. (7k � 3)(7k � 3) 14. (4d � 7)(4d � 7) 15. (3g � 9h)(3g � 9h)

16. (4q � 5t)(4q � 5t) 17. (a � 6u)2 18. (5r � s)2

19. (6c � m)2 20. (k � 6y)2 21. (u � 7p)2

22. (4b � 7v)2 23. (6n � 4p)2 24. (5q � 6s)2

25. (6a � 7b)(6a � 7b) 26. (8h � 3d)(8h � 3d) 27. (9x � 2y2)2

28. (3p3 � 2m)2 29. (5a2 � 2b)2 30. (4m3 � 2t)2

31. (6e3 � c)2 32. (2b2 � g)(2b2 � g) 33. (2v2 � 3e2)(2v2 � 3e2)

34. GEOMETRY Janelle wants to enlarge a square graph that she has made so that a sideof the new graph will be 1 inch more than twice the original side s. What trinomialrepresents the area of the enlarged graph?

GENETICS For Exercises 35 and 36, use the following information.In a guinea pig, pure black hair coloring B is dominant over pure white coloring b. Supposetwo hybrid Bb guinea pigs, with black hair coloring, are bred.

35. Find an expression for the genetic make-up of the guinea pig offspring.

36. What is the probability that two hybrid guinea pigs with black hair coloring will producea guinea pig with white hair coloring?

Practice Special Products

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8-88-8

Reading to Learn MathematicsSpecial Products

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8-88-8


Less

on

8-8

Pre-Activity When is the product of two binomials also a binomial?

Read the introduction to Lesson 8-8 at the top of page 458 in your textook.

What is a meant by the term trinomial product?

Reading the Lesson

1. Refer to the Key Concepts boxes on pages 458, 459, and 460.

a. When multiplying two binomials, there are three special products. What are the threespecial products that may result when multiplying two binomials?

b. Explain what is meant by the name of each special product.

c. Use the examples in the Key Concepts boxes to complete the table.

Symbols Product Example Product

Square of a Sum

Square of a Difference

Product of a Sum and a Difference

2. What is another phrase that describes the product of the sum and difference of two terms?


3. Explain how FOIL can help you remember how many terms are in the special productsstudied in this lesson.


Sums and Differences of Cubes Recall the formulas for finding some special products:

Perfect-square trinomials: (a � b)2 � a2 � 2ab � b2 or (a � b)2 � a2 � 2ab � b2

Difference of two squares: (a � b)(a � b) � a2 � b2

A pattern also exists for finding the cube of a sum (a � b)3.

1. Find the product of (a � b)(a � b)(a � b).

2. Use the pattern from Exercise 1 to evaluate (x � 2)3.

3. Based on your answer to Exercise 1, predict the pattern for the cube of a difference (a � b)3.

4. Find the product of (a � b)(a � b)(a � b) and compare it to your answer for Exercise 3.

5. Use the pattern from Exercise 4 to evaluate (x � 4)3.

Find each product.

6. (x � 6)3 7. (x � 10)3

8. (3x � y)3 9. (2x � y)3

10. (4x � 3y)3 11. (5x � 2)3

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-88-8

Chapter 8 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

ent

Write the letter for the correct answer in the blank at the right of each question.

1. Simplify y5 � y3.A. y2 B. y8 C. y15 D. 2y8 1.

2. Simplify (6x3)(x2).A. 6x5 B. 6x6 C. 7x5 D. 7x6 2.

3. Simplify (b4)3.A. b7 B. 3b4 C. b12 D. 3b7 3.

4. Simplify �aa

7

4�. Assume the denominator is not equal to zero.

A. a11 B. a28 C. a3 D. 1 4.

5. Simplify (5x2)(52x3).A. 25x6 B. 25x5 C. 125x6 D. 125x5 5.

6. Simplify ��2dc��3

. Assume the denominator is not equal to zero.

A. �6dc3

6� B. �

8dc� C. �

2d4

3c3� D. �

8dc3

3� 6.

7. Simplify �mm5

2nn

2


A. m7n5 B. �mn

3� C. m3n D. �m

n3� 7.

8. Simplify (x2)(x�3)(x�2).

A. x�12 B. x7 C. �x13� D. �x

1�3� 8.

9. Express 3851 in scientific notation.A. 3.851 � 103 B. 38.51 � 102 C. 385.1 � 10 D. 3.851 � 10�3 9.

10. Express 5.9 � 103 in standard notation.A. 5900 B. 0.0059 C. 59,000 D. 0.00059 10.

11. Evaluate (3 � 104)(3 � 105).A. 6 � 109 B. 9 � 109 C. 6 � 1020 D. 9 � 1020 11.

12. Find the degree of the polynomial b5 � 2b3 � 7.A. 3 B. 8 C. 5 D. 7 12.

88


Chapter 8 Test, Form 1 (continued)

13. Which of the following polynomials shows the terms of x2 � 5x3 � 4 � 2xarranged so that the powers of x are in descending order?A. 5x3 � 2x � x2 � 4 B. �4 � 2x � x2 � 5x3

C. 5x3 � 4 � 2x � x2 D. 5x3 � x2 � 2x � 4 13.

14. MONEY Write a polynomial to represent the value of d dimes and nnickels.A. 10d � 5n B. 0.1d � 0.5n C. 0.1d � 0.05n D. d � n 14.

15. Find (n2 � 3n) � (2n2 � n).A. 3n2 � 2n B. 3n2 � 2n C. n2 � 4n D. n2 � 2n 15.

16. Find (2a � 5) � (3a � 1).A. 5a � 6 B. a � 4 C. �a � 6 D. �a � 4 16.

17. Find 3m2(2m2 � m).A. 5m4 � 3m3 B. 6m4 � 3m2 C. 5m4 � 3m D. 6m4 � 3m3 17.

18. Simplify 3(x2 � 2x) � x(x � 1).A. 4x2 � x B. 2x2 � 7x C. 2x2 � 3x D. 2x2 � 5x 18.

19. Solve 3(2n � 6) � �4(n � 3).

A. 3 B. �35� C. 6 D. 1�

45� 19.

20. Find (x � 3)(x � 5).A. x2 � 8x � 15 B. x2 � 15 C. 2x � 8 D. 2x � 15 20.

21. Find (2n � 3)(n � 4).A. 3n � 1 B. 2n2 � 12C. 2n2 � 5n � 12 D. 2n2 � 11n � 1 21.

22. Find (x � 3)(2x2 � 4x � 8).A. 2x3 � 10x2 � 20x � 24 B. 4x2 � 4x � 24C. 12x2 � 20x � 24 D. 2x3 � 2x2 � 4x � 24 22.

23. Find (y � 5)2.A. y2 � 25 B. 2y � 10 C. y2 � 10y � 10 D. y2 � 10y � 25 23.

24. Find (3y � 1)2.A. 6y2 � 6y � 1 B. 9y2 � 6y � 1 C. 9y2 � 3y � 1 D. 9y2 � 6y � 1 24.

25. Find (2x � 5)(2x � 5).A. 4x B. 4x2 � 25 C. 4x2 � 20x � 25 D. 4x2 � 25 25.

Bonus Simplify (3n�1)(32n)4. B:

NAME DATE PERIOD

88

Chapter 8 Test, Form 2A

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SCORE


Ass

essm

ent


1. Simplify (9d4)(�6d5).A. 3d9 B. 3d C. �54d9 D. �54d20 1.

2. Simplify (x3)8.A. x24 B. x11 C. 8x24 D. 8x11 2.

3. Simplify (�2hk)4(4h3k5)2.A. 2h24k40 B. �64h9k11 C. � 256h10k14 D. 256h10k14 3.

4. Simplify �aa

9


A. a3 B. a12 C. a6 D. a27 4.

5. Simplify �93b6

�b4

1cc2


A. �27

c3b4� B. �

4cb3

4� C. �

27c3b3� D. �

4cb3

5� 5.

6. Simplify �((y3

2yn4n

�

6

3))2


A. �y916� B. �n

924� C. 9y16 D. 9n24 6.

7. Express 4173 in scientific notation.A. 4.173 � 103 B. 41.73 � 102 C. 417.3 � 10 D. 4.173 � 10�3 7.

8. Evaluate (4 � 10�2)(2 � 108).A. 8 � 10�16 B. 8 � 106 C. 6 � 106 D. 6 � 10�16 8.

9. Find the degree of the polynomial 3xy � 8x2y5 � x7y.A. 2 B. 7 C. 8 D. 10 9.

10. Arrange the terms of 4x3y2 � 6xy3 � 2x5 � 3y so that the powers of x are in descending order.A. 3y � 6xy3 � 4x3y2 � 2x5 B. 4x3y2 � 3y � 2x5 � 6xy3

C. 2x5 � 4x3y2 � 6xy3 � 3y D. �6xy3 � 4x3y2 � 3y � 2x5 10.

11. Find (9t2 � 4t � 6) � (t2 � 2t � 4).A. 8t2 � 6t � 10 B. 8t2 � 2t � 2 C. 9t2 � 6t � 2 D. 9t2 � 6t � 10 11.

88


Chapter 8 Test, Form 2A (continued)

12. CARS The number of domestic car sales from 1981–1999 in millions is modeled by the expression y � �0.01n2 � 0.11n � 6.51 where n is the number of years since 1981. The number of import car sales from 1981–1999 in millions is modeled by the expression y � 0.02n � 2.57. Find an expression that models the total number of car sales n years since 1981.A. �0.01n2 � 0.09n � 3.94 B. 0.01n2 � 0.13n � 9.08C. �0.01n2 � 0.31n � 8.08 D. �0.01n2 � 0.13n � 9.08 12.

13. Simplify 2a2(5a � 6) � 5a(a2 � 3a � 4) � 7(a � 5).A. 5a3 � 3a2 � 27a � 35 B. 5a3 � 27a2 � 13a � 35C. 5a3 � 10a � 7 D. none of these 13.

14. Find (c � 5)(c � 7).A. c2 � 12c � 35 B. c2 � 12c � 35C. c2 � 12c � 35 D. c2 � 35 14.

15. Find (3y � 4)(2y2 � y � 1).A. 6y3 � 5y2 � 7y � 4 B. 6y3 � 5y2 � 7y � 4C. 6y3 � 7y2 � 7y � 4 D. 6y3 � 5y2 � 7y � 4 15.

16. Find (3a � 2b)(3a � 2b).A. 9a2 � 4b2 B. 9a2 � 4b2

C. 9a2 � 12ab � 4b2 D. 9a2 � 12ab � 4b2 16.

17. Find (4a2 � b)2.A. 16a4 � b2 B. 8a4 � b2

C. 16a4 � 8a2b � b2 D. 4a4 � 8a2b � b2 17.

18. Solve 6(n � 11) � 12 � 4(2n � 3).A. �11 B. 11 C. �33 D. 33 18.

19. Solve 5x2 � 3x � (7x2 � 5x) � (2x2 � 16).A. 2 B. �2 C. 8 D. �8 19.

20. GEOMETRY The length of a rectangle is 4 units less than twice the width.If the length is decreased by 3 units and the width is increased by 1 unit,the area is decreased by 16 square units. If w is the original width, which equation must be true?A. (2w � 3)(w � 3) � w(2w � 4) � 16B. 2(2w � 3) � 2(w � 3) � 2w � 2(2w � 4) � 16C. (2w � 7)(w � 1) � w(2w � 4) � 16D. 2(2w � 7) � 2(w � 1) � 2w � (2w � 4) � 16 20.

Bonus Simplify �773

x

x

�

�

3

1�. B:

NAME DATE PERIOD

88

Chapter 8 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Simplify (5d3)( � 2d2).A. 10d5 B. �10d5 C. 10d6 D. �10d6 1.

2. Simplify (m4)2.A. 6m B. m8 C. m6 D. 2m4 2.

3. Simplify (�2xy2)4(2x3y4)2.A. 4x24y32 B. �8x9y6 C. 64x10y16 D. �4x10y16 3.

4. Simplify �aa1

4


A. a3 B. a16 C. a48 D. a8 4.

5. Simplify �26nn�

�

1y3

�y


A. �4ny2

3� B. �

3ny2

4� C. �n4

3y2� D. �

3yn4

2� 5.

6. Simplify �((aa�

4

2

bb�

4

8))�

3


A. ab3 B. 1 C. �ab4

2

8

4� D. �a

b4

2

8

4� 6.

7. Express 5.43 � 10�4 in standard notation.A. 54,300 B. 0.00543 C. 5430 D. 0.000543 7.

8. Evaluate �22..488

��10

1�03

7�.

A. 0.48 � 10�4 B. 1.2 � 10�4 C. 1.2 � 1010 D. 0.48 � 1010 8.

9. Find the degree of the polynomial 2x3y � 4xy2 � 9x3y2.A. 4 B. 3 C. 12 D. 5 9.

10. Arrange the terms of x2y3 � 4xy2 � 3x3y � 6 so that the powers of x are in ascending order.A. 6 � 4xy2 � x2y3 � 3x3y B. x2y3 � 3x3y � 4xy2 � 6C. 6 � 4xy2 � 3x3y � x2y3 D. 6 � 3x3y � 4xy2 � x2y3 10.

11. Find (3c2 � 8c � 5) � (c2 � 8c � 6).A. 3c2 � 1 B. 4c2 � 11 C. 4c2 � 16c � 1 D. 2c2 � 16c � 1 11.

88


Chapter 8 Test, Form 2B (continued)

12. CLUBS The number of girls in a local 4-H club is modeled by the expression 4n2 � n � 52 where n is the number of years after 1990. The number of girls in 4H who are age 8 and under is modeled by the expression n2 � 13.Find an expression that models the number of girls older than 8 in the club.A. 3n2 � n � 39 B. 3n2 � n � 39C. �3n2 � n � 39 D. �3n2 � n � 39 12.

13. Simplify 3b2(4b � 7) � 2b(b2 � 5b � 3) � 6(b � 2).A. 14b3 � 11b2 � 12b � 12 B. 41b2 � 12C. 14b3 � 31b2 � 12b � 12 D. 10b3 � 31b2 � 12 13.

14. Find (x � 2)(x � 4).A. x2 � 8 B. x2 � 2x � 6C. x2 � 2x � 8 D. x2 � 6x � 8 14.

15. Find (3x � 2)(4x2 � 2x � 7).A. 12x3 � 2x2 � 25x � 14 B. 12x3 � 14x2 � 25x � 14C. 7x3 � 9x2 � 25x � 14 D. 7x3 � 7x2 � 4x � 5 15.

16. Find (3y � 4z)(3y � 4z).A. 9y2 � 16z2 B. 9y2 � 24yz � 16z2

C. 9y2 � 16z2 D. 9y2 � 24yz � 16z2 16.

17. Find (�2r2 � s)2.A. 4r4 � s2 B. 4r4 � 4r2s � s2

C. �4r4 � s2 D. �4r4 � 4r2s � s2 17.

18. Solve �4(5 � 2n) � 8( � 6 � 5n).

A. ��19� B. ��

238� C. ��

78� D. ��1

72�

18.

19. Solve x(x � 3) � 2 � 2 � x(x � 1).A. 2 B. �2 C. 1 D. 0 19.

20. ART A picture is 4 inches longer than it is wide. It is surrounded by a mat that is 2 inches wide. The total area of the mat is 112 square inches. If wis the width of the picture, which equation is true?A. (w � 4)(w � 8) � w(w � 4) � 112B. (w � 2)(w � 6) � w(w � 4) � 112C. (w � 4)(w � 8) � w(w � 4) � 112D. (w � 2)(w � 6) � w(w � 4) � 112 20.

Bonus Simplify 32n�1 � 35n. B:

NAME DATE PERIOD

88

Chapter 8 Test, Form 2C


Simplify.

1. 3y5 � y3 1.

2. (9m3n5)(�2mn2) 2.

3. (w5y4)3 3.

4. 4a3n6 � 4(a3n)6 � 4(an2)3 4.

For Questions 5–7, simplify. Assume that no denominator is equal to zero.

5. �pp6

3qq2

� 5.

6. �146rr�

3

1ss�

2

5� 6.

7. �(�

(48xx3

2

yy)2

3)2

� 7.

8. Express 0.000498 in scientific notation. 8.

9. Express 1.27 � 105 in standard notation. 9.

For Questions 10 and 11, express each result in scientific and standard notation.

10. Evaluate (2.5 � 10�2)(4 � 106). 10.

11. The radius of Earth is approximately 2.51 � 108 inches. The 11.radius of the Sun is approximately 2.74 � 1010 inches. How many times greater is the radius of the Sun than the radius of Earth?

12. Find the degree of the polynomial 2x3y3 � 4xy � 10x3y. 12.

13. Arrange the terms of the polynomial 4 � 3x3y3 � x5y � xy so 13.that the powers of x are in descending order.

NAME DATE PERIOD

SCORE 88

Ass

essm

ent


Chapter 8 Test, Form 2C (continued)


14. (5n2 � 2ny � 3y2) � (9n2 � 8ny � 10y2) 14.

15. (11m2 � 2mn � 8n2) � (8m2 � 4mn � 2n2) 15.

16. (x2 � 5y) � (2x2 � 6y) 16.

Find each product.

17. 5hk2(2h2k � hk3 � 4h2k2) 17.

18. (4x2 � 2y2)(2x2 � y2) 18.

19. (3s � 5)(2s2 � 8s � 6) 19.

20. (5c � 4)2 20.

21. (7a � 3b)(7a � 3b) 21.

22. (4n � 1)2 22.

For Questions 23 and 24, solve each equation.

23. �6(3n � 2) � 4(�3 � 2n) 23.

24. 8n � 11 � 4 � 5(2n � 1) 24.

25. GARDENING The length of a rectangular garden is 8 feet 25.longer than the width. The garden is surrounded by a 4-foot sidewalk. The sidewalk has an area of 320 square feet.Find the dimensions of the garden.

Bonus If you multiply (x � 1)20, how many terms will there B:be? (Hint: Look for a pattern in the smaller powers of (x � 1).)

NAME DATE PERIOD

88

Chapter 8 Test, Form 2D


Simplify.

1. 5x4 � x3 1.

2. (3a2b5)(�2ab3) 2.

3. (w3z7)3 3.

4. 4a4b8 � 2(ab2)4 � 4(a2b4)2 4.


5. ��

41m2m

5

3� 5.

6. �48bb

�

2d3

�d2

5� 6.

7. �(�(3

3rr3

2ss5

7)3

)2� 7.

8. Express 12,556 in scientific notation. 8.

9. Express 7.43 � 10�3 in standard notation. 9.


10. The minimum distance from Earth to the Moon is 10.approximately 2.26 � 105 miles. There are approximately 6.34 � 104 inches in one mile. What is the minimum distance from Earth to the Moon in inches?

11. �25.7.4

��

1100�

4

3� 11.

12. Find the degree of the polynomial 2x2y � 4x5 � 6xy3. 12.

13. Arrange the terms of the polynomial 3x2y � xy2 � 3y3 � x3 13.so that the powers of x are in descending order.

NAME DATE PERIOD

SCORE 88

Ass

essm

ent


Chapter 8 Test, Form 2D (continued)


14. (7m2 � 3m � 4) � (3m2 � 9m � 5) 14.

15. (4y2 � 3y � 7) � (4y2 � 7y � 2) 15.

16. (3a3 � 4b) � (2a3 � b) 16.

Find each product.

17. 3x2y(2x2y � 5xy2 � 8y3x2) 17.

18. (3r2 � 5s2)(3r2 � 5s2) 18.

19. (2n � 3)(3n2 � 4n � 1) 19.

20. (5y � 6)2 20.

21. (2k � 5r)(2k � 5r) 21.

22. (2c � 1)2 22.

For Questions 23 and 24, solve each equation.

23. 5x � 8 � 3 � 2(3x � 4) 23.

24. �5(2n � 3) � 7(3 � n) 24.

25. GARDENING The length of a rectangular garden is 5 feet 25.longer than its width. The garden is surrounded by a 2-foot-wide sidewalk. The sidewalk has an area of 76 square feet. Find the dimensions of the garden.

Bonus If (x � 1)10 is multiplied out, how many terms will there B:be? (Hint: Look for a pattern in the smaller powers of (x � 1).)

NAME DATE PERIOD

88

Chapter 8 Test, Form 3


Simplify.

1. (ab8)(3a6b2) 1.

2. ��23�h3�4

2.

3. ��23�r�3

(27r)(5s)2��12�s4�3

3.

4. (43x�7)(42x�9) 4.


5. ��

95c46cd2

2d5

� 5.

6. �(�82mm

�x�

5x3

0)�4

� 6.

7. ��63aa3

2

bb�

�

4

3��2��4�a

5�b3��3

7.

8. Express 196,783 in scientific notation. 8.


9. (7.2 � 10�3)(8.1 � 10�2) 9.

10. �4.55.91

��

1100

�

2

3� 10.

11. Arrange the terms of the polynomial 2xy � 6 � 4x5y2 � 7x6y3 11.so that the powers of x are in descending order.

12. Find the degree of the polynomial m2np2 � mn3p2 � 4m4n. 12.

NAME DATE PERIOD

SCORE 88

Ass

essm

ent


Chapter 8 Test, Form 3 (continued)

For Questions 13 and 14, find each sum or difference.

13. (8w2 � 4w � 2) � (2w2 � w � 6) 13.

14. (7u2v � 3uv � 4uv2) � (4uv � 3u2v � 2uv2) 14.

15. GEOMETRY The measures of two sides 15.of a triangle are given on the triangle at the right. If the perimeter of the triangle is 6x2 � 8y, find the measure of the third side.

16. Simplify 5n2(n � 6) � 2n(3n2 � n � 6) � 7(n2 � 3). 16.

For Questions 17–20, find each product.

17. (2y � 7)(4y � 4) 17.

18. ��23�m � 1��

12�m � 2� 18.

19. (4x � y)(2x2 � xy � 5y2) 19.

20. (5r2 � 3s2)(5r2 � 3s2) 20.

21. A square has sides of length (3x � 1) feet. Write an 21.expression that represents the area of the square.

22. Solve y(y � 6) � (5y2 � 36) � (4y2 � 3y). 22.

23. If f(x) � x2 � 5x and g(x) � 2x � x2, find f(a � 1) � g(a � 1). 23.

24. If a2 � b2 � 11 and ab � 3, find the value of (a � b)2. 24.

25. GEOMETRY The length of a rectangle is 4 centimeters 25.more than its width. If the length is increased by 8 centimeters and the width is decreased by 4 centimeters,the area will remain unchanged. Find the original dimensions of the rectangle.

Bonus Graph the solution set of B:(x � 3)(x � 5) � (x � 1)2 � 2(x � 1).

NAME DATE PERIOD

88

x2 � y

2x2 � x � 5y

Chapter 8 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.

1. For each equation determine why the statement is incorrect.Explain how to change the right side of the equation to make a true statement.a. (4u � 5v) � (u � v) � 3u � 4vb. x2y(3x3 � 4) � 3x6y � 4x2yc. (3a � 5b)2 � 9a2 � 25b2

2. a. State whether the number 23.4 � 108 is in scientific notation.Explain how you determined your answer.

b. Explain why scientific notation is helpful when dividing22,100,000 by 0.00000013. Find the quotient 22,100,000 0.00000013.

3. Give a counterexample for each statement.a. The degree of a polynomial is always 2 or greater.b. The degree of a trinomial is always greater than the degree

of a monomial.

4. a. Simplify ��24aa�3

2��4by using the Quotient of Powers property

first, then use the Power of a Power property.

b. Simplify ��24aa�3

2��4by using the Power of a Quotient property

first, then use the Quotient of Powers property.c. Write a statement that generalizes the results of part a

and part b.

5. Draw an area model demonstrating the product (3x � y)(x � 2y).Find the product algebraically. Explain how the algebraic productverifies the area model.

NAME DATE PERIOD

SCORE 88

Ass

essm

ent


Chapter 8 Vocabulary Test/Review

Write the letter of the term that best matches each statement or phrase.

1. constant

2. Power of a Power

3. Product of Powers

4. FOIL method

5. Quotient of Powers

6. Power of a Quotient

7. monomial

8. zero exponent

9. trinomial

10. binomial

In your own words—Define each term.

11. degree of a monomial

12. degree of a polynomial

13. scientific notation

binomialconstantdegree of a monomialdegree of a polynomialdifference of squaresFOIL method

monomialnegative exponentpolynomialPower of a PowerPower of a ProductProduct of Powers

Power of a QuotientQuotient of Powersscientific notationtrinomialzero exponent

NAME DATE PERIOD

SCORE 88

a. property that tells us to find the power ofthe numerator and the power of thedenominator

b. property that tells us that any nonzeronumber raised to the zero power is 1

c. property that tells us to multiplyexponents

d. the sum of three monomials

e. monomials that are real numbers

f. used to multiply two binomials

g. property that tell us to add exponentswhen multiplying two powers that havethe same base

h. the sum of two monomials

i. property that tells us to subtractexponents when dividing two powers thathave the same base

j. a number, a variable, or a product of anumber and one or more variables

Chapter 8 Quiz (Lessons 8–1 and 8–2)

88


Simplify.

1. (r3)(2r5) 2. (x5)4

3. (�4m2n3)(3mn4) 4. (�5x4y2)3

5. (2cd)2(�4c3)2 6. (5y2w4)2 � 2(yw2)4


7. �661

9

5� 8. �

yy

8

3� 9. �rr6

4nn�

2

7�

10. STANDARDIZED TEST PRACTICE Write the ratio of the area of a circle with radius r to the circumference of the same circle.

A. �2r� B. 2 C. �2

r� D. �2

1r�

10.

NAME DATE PERIOD

SCORE


For Questions 1 and 2, express each number in scientific notation.

1. 57,600 2. 0.0000061

3. Express 6.4871 � 10�3 in standard notation.


4. �74.2.5

��

1100�

2

5� 5. (3.5 � 106)(8.2 � 103)


6. 5a � 2b2 � 1 7. 24xy � xy3 � x2

For Questions 8 and 9, arrange the terms of each polynomial so that the powers of x are in descending order.

8. 4x2 � 3x3 � 2x � 12 8.

9. 5x3y � 3xy4 � x2y3 � y4 9.

10. Arrange the terms of the polynomial �3 � x2 � 4x so that 10.the powers of x are in ascending order.

NAME DATE PERIOD

SCORE 88

Ass

essm

ent

1.

2.

3.

4.

5.

6.

7.

8.

9.

1.

2.

3.

4.

5.

6.

7.


For Questions 1 and 2, find each sum or difference.

1. (�2x2 � x � 6) � (5x2 � 4x � 2) 1.

2. (5a � 9b) � (2a � 4b) 2.

3. Find 3xy(2x2 � 5xy � 7y2). 3.

4. Simplify 5c2(c � 10) � 4c(2c2 � 6c � 1). 4.

5. Solve 3(2p � 1) � 7(5 � p) � 6. 5.

Chapter 8 Quiz (Lesson 8–7)

Find each product.

1. (4m � 1)(m � 2) 1.

2. (2c � 3)(c2 � 4c � 5) 2.

3. (4h � 3)2 3.

4. (a � 9)2 4.

5. (2x � 6y)(2x � 6y) 5.

6. (m2 � 2n)2 6.

7. (9x � 7)(9x � 7) 7.

8. (a � 3b)(a � 3b) 8.

For Exercises 9 and 10, use the figure.

9. Write an expression to represent the area of the top side of 9.the prism.

10. Write an expression to represent the volume of the prism. 10.

4a � 4a � 1

a � 3

NAME DATE PERIOD

SCORE


88

NAME DATE PERIOD

SCORE

88

Chapter 8 Mid-Chapter Test (Lessons 8–1 through 8–4)



1. Simplify (n5)(n2)(m3)(m4).A. n10m12 B. n7m7 C. nm7 D. nm14 1.

2. Simplify (3w2v)2( � 2w5v2)3.A. �72w19v8 B. 72w12v7 C. �36w32v10 D. 36w19v6 2.

For Questions 3 and 4, simplify. Assume the denominator is not equal to zero.

3. �mm

6

2nn

3

6�

A. �mn3

4� B. ��

mn3

4� C. ��

mn3

8� D. �

mn3

8� 3.

4. �((zz2

3ww

�

2

1

))2

3�

A. �w1

7� B. �zw1

7

2� C. w D. �w

1� 4.

5. Express 31,000,000 in scientific notation.A. 31 � 106 B. 3.1 � 10�7

C. 3.1 � 107 D. 31 � 10�6 5.

6. Find the degree of the polynomial 4x2y3 � 2xy2 � 5x3y.A. 4 B. 3 C. 6 D. 5 6.

7. Express 4.02 � 10�3 in standard notation.A. 0.00402 B. 402,000 C. 4020 D. 402 7.


8. �15.1

�7 �

101�01

2� 9. (8.3 � 102)(9.1 � 10�7)

For Questions 10 and 11, simplify. Assume that no denominator is equal to zero.

10. �52(xx4

3

yy2

5

)3� 11. ��1

�41m0m

�7

�

y1

�y3

0

rr�4�

12. Find the degree of 5c2 � 3c4 � 2c3 � 1. 12.

Arrange the terms of each polynomial so that the powers of x are in descending order. 13.

13. 6 � 2x2 � 3x � 4x3

14. 2a2x3 � 8x4 � 3a4x2 � 8x 14.

Part II

Part I

NAME DATE PERIOD

SCORE 88

Ass

essm

ent

8.

9.

10.

11.


Chapter 8 Cumulative Review (Chapters 1–8)

1. Find a counterexample for the conditional statement. 1.If a number is divisible by 5, then the one’s digit is a 5.(Lesson 1-7)

2. Solve a(b � c) � �ab� for c. (Lesson 3-8) 2.

3. Write an equation in function 3.notation for the relation.(Lesson 4-8)

4. TUITION The average cost of tuition and fees at a four-year 4.college was $3,356 for the year 1999–2000. For the year 1997–1998 the average cost was $3,111. Find the rate of change between the year 1997–1998 and the year 1999–2000. (Lesson 5-1)

5. Write y � 1 � 4(x � 2) in slope intercept form. (Lesson 5-5) 5.

6. Solve 14u � 9 � 10u 21. (Lesson 6-3) 6.

7. Solve 3y � 4 � 7. (Lesson 6-5) 7.

8. RECREATION Barrington needs to buy snorkels and fins for 8.his family for their annual beach vacation. Snorkels cost $8 a set and fins cost $12 a pair. Write an inequality that represents this situation if Barrington has $62 to spend. (Lesson 6-6)

9. Graph the system of equations. Then determine whether 9.

the system has no solution, one solution, or infinitely manysolutions. If the system has one solution, name it. (Lesson 7-1)

x � y � 4y � x

10. Determine the best method to solve the system of equations.Then solve the system. (Lesson 7-4)

5x � 2y � 72x � 5y � �3 10.

11. Simplify �5117x�

x2

1

yy3

�. Assume the denominator is not equal to

zero. (Lesson 8-2) 11.

12. Evaluate (6 � 105)(2.3 � 103). Express the result in 12.scientific and standard notation. (Lesson 8-3)

13. Find ��23�a2 � �

14�a � �

15��

13�a2 � �

34�a � �

25��. (Lesson 8-5) 13.

14. Find (3x � 2)2. (Lesson 8-8) 14.

y

xO

NAME DATE PERIOD

88

y

xO

Standardized Test Practice (Chapters 1–8)


1. Write y � 11 � �12�(x � 12) in standard form. (Lesson 5-5)

A. y � �12�x � 5 B. y � �

12�x � 17 C. x � 2y � �10 D. x � 2y � 17 1.

2. Ashanti wants to collect more than 50 food items for the local homeless shelter. If he has already collected 13, how many more items must he collect? (Lesson 6-1)

E. 37 F. at least 38 G. at least 37 H. more than 36 2.

3. Solve �5n � 22 � �73. (Lesson 6-3)

A. {n � n � 19} B. {n � n � 19} C. �n � n � ��551�� D. {n � n � 90} 3.

4. Solve � 5k � 2 � 1. (Lesson 6-5)

E. �k � k � ��35� or k ��

15�� F. �k � k � �

35� � k � �

15��

G. H. {k � k is a real number.} 4.

5. Use substitution to solve the system of equations. (Lesson 7-2)

y � �2x5x � 3y � 4A. (�4, 8) B. (8, �4) C. (�8, 4) D. (4, �8) 5.

6. Use elimination to solve the system of equations. (Lesson 7-3)

x � 3y � �62x � 3y � �9E. (�2, �4) F. (�6, 1) G. (�3, �1) H. (3, �5) 6.

7. Simplify �(3(�32

2))((33

3

�)3)�. (Lesson 8-2)

A. 310 B. 312 C. �1 D. �13� 7.

8. Express 0.46 � 103 in scientific notation. (Lesson 8-3)

E. 0.46 � 103 F. 4.6 � 102 G. 4.6 � 103 H. 46 � 101 8.

9. Find (3n2 � 6n) � (5n3 � 2n) � (2n3 � n2). (Lesson 8-5)

A. 3n3 � 2n2 � 8n B. 6n3 � n2 � 4nC. 3n3 � 4n2 � 4n D. 6n3 � 4n2 � 8n 9.

10. Find (7b � 11)(b � 3). (Lesson 8-7)

E. 7b2 � 10b � 33 F. 7b2 � 11b � 33G. 7b2 � 11b � 8 H. 7b2 � 10b � 8 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

88

Ass

essm

ent

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.


Standardized Test Practice (continued)

11. Evaluate 4x(y2 � z2) if x � 3, y � 5 and z � 4. 11. 12.(Lesson 1-2)

12. After receiving his allowance, Spencer spent half of his money on a card for his mother.He bought 2 toy cars for $0.98 to give to his brothers, and a soft drink for $0.35. How much did Spencer receive for his allowance if he has $0.42 left over? (Lesson 3-4)

13. At the end of the 2001 football season, the 13. 14.Dallas Cowboys and the Denver Broncos had won a total of 7 Super Bowls. The Cowboys had won 2.5 times as many Super Bowls as the Broncos. How many Super Bowls had the Dallas Cowboys won? (Lesson 7-2)

14. Express 5.7 � 10�2 in standard notation.(Lesson 8-3)

Column A Column B

15. 15.

(Lesson 2-7)

16. y � 3x � 10 16.y � �2x � 15.

(Lesson 7-2)

17. 17.

(Lesson 8-4)

DCBAthe degree of3x2y2 � 6x2y3 � 8x3y2

the degree of7x5 � 4x3 � x6

yx

DCBA

DCBA

��191��

57

��

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

NAME DATE PERIOD

88

NAME DATE PERIOD

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

A

D

C

B

Standardized Test PracticeStudent Record Sheet (Use with pages 470–471 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 1

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

For Questions 11 and 13, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

11 (grid in) 11 13

12

13 (grid in)

14

15

16

Select the best answer from the choices given and fill in the corresponding oval.

17 21

18 22

19 23

20

Record your answers for Question 24 on the back of this paper.

DCBA

DCBADCBA

DCBADCBA

DCBADCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBA

DCBADCBADCBADCBA

DCBADCBADCBADCBA

NAME DATE PERIOD

88

An

swer

s

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 4 Open-EndedPart 4 Open-Ended

Part 1 Multiple ChoicePart 1 Multiple Choice


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er o

f a

Pro

duct

�(�

2)3 (

a3)(

a8)(

b2)3

Com

mut

ativ

e P

rope

rty

�(�

2)3 (

a11 )

(b2 )

3P

rodu

ct o

f P

ower

s

��

8a11

b6P

ower

of

a P

ower

Th

e pr

odu

ct i

s �

8a11

b6.

Sim

pli

fy.

1.(y

5 )2

2.(n

7 )4

3.(x

2 )5 (

x3)

y10

n28

x13

4.�

3(ab

4 )3

5.(�

3ab4

)36.

(4x2

b)3

�3a

3 b12

�27

a3 b

1264

x6 b

3

7.(4

a2)2

(b3 )

8.(4

x)2 (

b3)

9.(x

2y4

)5

16a4

b3

16x

2 b3

x10 y

20

10.(

2a3 b

2 )(b

3 )2

11.(

�4x

y)3 (

�2x

2 )3

12.(

�3j

2 k3 )

2 (2j

2 k)3

2a3 b

851

2x9 y

372

j10 k

9

13.(

25a2

b)3 �

abc �2

14.(

2xy)

2 (�

3x2 )

(4y4

)15

.(2x

3 y2 z

2 )3 (

x2z)

4

625a

8 b5 c

2�

48x

4 y6

8x17

y6 z

10

16.(

�2n

6 y5 )

(�6n

3 y2 )

(ny)

317

.(�

3a3 n

4 )(�

3a3 n

)418

.�3(

2x)4

(4x5

y)2

12n

12y1

0�

243a

15n

8�

768x

14y

2

1 � 5Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Mu

ltip

lyin

g M

on

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 8-1)


An

swer

s

Skil

ls P

ract

ice

Mu

ltip

lyin

g M

on

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

©G

lenc

oe/M

cGra

w-H

ill45

7G

lenc

oe A

lgeb

ra 1

Lesson 8-1

Det

erm

ine

wh

eth

er e

ach

exp

ress

ion

is

a m

onom

ial.

Wri

te y

esor

no.

Exp

lain

.

1.11

Yes;

11 is

a r

eal n

um

ber

an

d a

n e

xam

ple

of

a co

nst

ant.

2.a

�b

No

;Th

is is

th

e d

iffe

ren

ce,n

ot

the

pro

du

ct,o

f tw

o v

aria

ble

s.

3.N

o;T

his

is t

he

qu

oti

ent,

no

t th

e p

rod

uct

,of

two

var

iab

les.

4.y

Yes;

Sin

gle

var

iab

les

are

mo

no

mia

ls.

5.j3

kYe

s;T

his

is t

he

pro

du

ct o

f tw

o v

aria

ble

s.

6.2a

�3b

No

;Th

is is

th

e su

m o

f tw

o m

on

om

ials

.

Sim

pli

fy.

7.a2

(a3 )

(a6 )

a11

8.x(

x2)(

x7)

x10

9.(y

2 z)(

yz2 )

y3z3

10.(

�2 k2 )

(�3 k

)�5 k

3

11.(

e2f4

)(e2

f2)

e4f6

12.(

cd2 )

(c3 d

2 )c4

d4

13.(

2x2 )

(3x5

)6x

714

.(5a

7 )(4

a2)

20a9

15.(

4xy3

)(3x

3 y5 )

12x

4 y8

16.(

7a5 b

2 )(a

2 b3 )

7a7 b

5

17.(

�5m

3 )(3

m8 )

�15

m11

18.(

�2c

4 d)(

�4c

d)

8c5 d

2

19.(

102 )

310

6o

r 1,

000,

000

20.(

p3)1

2p

36

21.(

�6p

)236

p2

22.(

�3y

)3�

27y

3

23.(

3pq2

)29p

2 q4

24.(

2b3 c

4 )2

4b6 c

8

GEO

MET

RYE

xpre

ss t

he

area

of

each

fig

ure

as

a m

onom

ial.

25.

26.

27.

x7c2

d2

18p

44p 9p3

cd

cdx2

x5

p2� q2

©G

lenc

oe/M

cGra

w-H

ill45

8G

lenc

oe A

lgeb

ra 1

Det

erm

ine

wh

eth

er e

ach

exp

ress

ion

is

a m

onom

ial.

Wri

te y

esor

no.

Exp

lain

.

1.N

o;

this

invo

lves

th

e q

uo

tien

t,n

ot

the

pro

du

ct,o

f va

riab

les.

2.Ye

s;th

is is

th

e p

rod

uct

of

a n

um

ber

,,a

nd

tw

o v

aria

ble

s.

Sim

pli

fy.

3.(�

5x2 y

)(3x

4 )�

15x

6 y4.

(2ab

2 c2 )

(4a3

b2c2

)8a

4 b4 c

4

5.(3

cd4 )

(�2c

2 )�

6c3 d

46.

(4g3

h)(

�2g

5 )�

8g8 h

7.(�

15xy

4 )��

xy3 �

5x2 y

78.

(�xy

)3(x

z)�

x4 y

3 z

9.(�

18m

2 n)2��

mn

2 ��

54m

5 n4

10.(

0.2a

2 b3 )

20.

04a4

b6

11. �

p �2p

212

. �cd

3 �2c2

d6

13.(

0.4k

3 )3

0.06

4k9

14.[

(42 )

2 ]2

48o

r 65

,536

GEO

MET

RYE

xpre

ss t

he

area

of

each

fig

ure

as

a m

onom

ial.

15.

16.

17.

18a

3 b6

(25x

6 )�

12a3

c4

GEO

MET

RYE

xpre

ss t

he

volu

me

of e

ach

sol

id a

s a

mon

omia

l.

18.

19.

20.

27h

6m

4 n5

(63g

4 )�

21.C

OU

NTI

NG

A p

anel

of

fou

r li

ght

swit

ches

can

be

set

in 2

4w

ays.

A p

anel

of

five

lig

ht

swit

ches

can

set

in

tw

ice

this

man

y w

ays.

In h

ow m

any

way

s ca

n f

ive

ligh

t sw

itch

es

be s

et?

25o

r 32

22.H

OB

BIE

ST

awa

wan

ts t

o in

crea

se h

er r

ock

coll

ecti

on b

y a

pow

er o

f th

ree

this

yea

r an

dth

en i

ncr

ease

it

agai

n b

y a

pow

er o

f tw

o n

ext

year

.If

she

has

2 r

ocks

now

,how

man

yro

cks

wil

l sh

e h

ave

afte

r th

e se

con

d ye

ar?

26o

r 64

7g2

3g

m3 n

mn3

n

3h2

3h2

3h2

6ac3

4a2 c

5x3

6a2 b

4

3ab2

1 � 161 � 4

4 � 92 � 3

1 � 6

1 � 3

1 � 2b3 c

2�

2

21a2

�7b

Pra

ctic

e (

Ave

rag

e)

Mu

ltip

lyin

g M

on

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1



Readin

g t

o L

earn

Math

em

ati

csM

ult

iply

ing

Mo

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1

©G

lenc

oe/M

cGra

w-H

ill45

9G

lenc

oe A

lgeb

ra 1

Lesson 8-1

Pre-

Act

ivit

yW

hy

doe

s d

oub

lin

g sp

eed

qu

adru

ple

bra

kin

g d

ista

nce

?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-1

at

the

top

of p

age

410

in y

our

text

book

.

Fin

d tw

o ex

ampl

es i

n t

he

tabl

e to

ver

ify

the

stat

emen

t th

at w

hen

spe

ed i

sdo

ubl

ed,t

he

brak

ing

dist

ance

is

quad

rupl

ed.W

rite

you

r ex

ampl

es i

n t

he

tabl

e.

Sp

eed

Bra

kin

g D

ista

nce

S

pee

d D

ou

ble

d

Bra

kin

g D

ista

nce

(m

iles

per

ho

ur)

(fee

t)(m

iles

per

ho

ur)

Qu

adru

ple

d (

feet

)

2020

4080

3045

6018

0

Rea

din

g t

he

Less

on

1.D

escr

ibe

the

expr

essi

on 3

xyu

sin

g th

e te

rms

mon

omia

l,co

nst

ant,

vari

able

,an

d pr

odu

ct.

Th

e m

on

om

ial 3

xyis

th

e p

rod

uct

of

the

con

stan

t 3

and

th

e va

riab

les

xan

d y

.

2.C

ompl

ete

the

char

t by

ch

oosi

ng

the

prop

erty

th

at c

an b

e u

sed

to s

impl

ify

each

expr

essi

on.T

hen

sim

plif

y th

e ex

pres

sion

.

Exp

ress

ion

Pro

per

tyE

xpre

ssio

n S

imp

lifie

d

Pro

duct

of

Pow

ers

35�

32P

ower

of

a P

ower

37o

r 21

87P

ower

of

a P

rodu

ct

Pro

duct

of

Pow

ers

(a3 )

4P

ower

of

a P

ower

a12

Pow

er o

f a

Pro

duct

Pro

duct

of

Pow

ers

(�4x

y)5

Pow

er o

f a

Pow

er

�10

24x

5 y5

Pow

er o

f a

Pro

duct

Hel

pin

g Y

ou

Rem

emb

er

3.W

rite

an

exa

mpl

e of

eac

h o

f th

e th

ree

prop

erti

es o

f po

wer

s di

scu

ssed

in

th

is l

esso

n.

Th

en,u

sin

g th

e ex

ampl

es,e

xpla

in h

ow t

he

prop

erty

is

use

d to

sim

plif

y th

em.

Sam

ple

an

swer

:F

or

z2�

z5 ,

sin

ce t

he

bas

es a

re t

he

sam

e,u

se t

he

Pro

du

ct o

f P

ow

ers

Pro

per

ty a

nd

ad

d t

he

exp

on

ents

to

get

z7 .

Fo

r (a

4 )3 ,

use

th

e P

ow

er o

f a

Po

wer

Pro

per

ty.M

ult

iply

th

e ex

po

nen

ts t

o g

et a

12.

Fo

r (3

rs)3

,use

th

e P

ow

er o

f a

Pro

du

ct P

rop

erty

.Rai

se t

he

con

stan

t an

dea

ch v

aria

ble

to

th

e p

ow

er t

o g

et 2

7 r3 s

3 .

©G

lenc

oe/M

cGra

w-H

ill46

0G

lenc

oe A

lgeb

ra 1

An

Wan

g

An

Wan

g (1

920–

1990

) w

as a

n A

sian

-Am

eric

an w

ho

beca

me

one

of t

he

pion

eers

of

the

com

pute

r in

dust

ry i

n t

he

Un

ited

Sta

tes.

He

grew

up

inS

han

ghai

,Ch

ina,

but

cam

e to

th

e U

nit

ed S

tate

s to

fu

rth

er h

is s

tudi

esin

sci

ence

.In

194

8,h

e in

ven

ted

a m

agn

etic

pu

lse

con

trol

lin

g de

vice

that

vas

tly

incr

ease

d th

e st

orag

e ca

paci

ty o

f co

mpu

ters

.He

late

rfo

un

ded

his

ow

n c

ompa

ny,

Wan

g L

abor

ator

ies,

and

beca

me

a le

ader

in

the

deve

lopm

ent

of d

eskt

op c

alcu

lato

rs a

nd

wor

d pr

oces

sin

g sy

stem

s.In

198

8,W

ang

was

ele

cted

to

the

Nat

ion

al I

nve

nto

rs H

all

of F

ame.

Dig

ital

com

pute

rs s

tore

in

form

atio

n a

s n

um

bers

.Bec

ause

th

eel

ectr

onic

cir

cuit

s of

a c

ompu

ter

can

exi

st i

n o

nly

on

e of

tw

o st

ates

,op

en o

r cl

osed

,th

e n

um

bers

th

at a

re s

tore

d ca

n c

onsi

st o

f on

ly t

wo

digi

ts,0

or

1.N

um

bers

wri

tten

usi

ng

only

th

ese

two

digi

ts a

re c

alle

db

inar

y n

um

ber

s.T

o fi

nd

the

deci

mal

val

ue

of a

bin

ary

nu

mbe

r,yo

uu

se t

he

digi

ts t

o w

rite

a p

olyn

omia

l in

2.F

or i

nst

ance

,th

is i

s h

ow t

ofi

nd

the

deci

mal

val

ue

of t

he

nu

mbe

r 10

0110

1 2.(

Th

e su

bscr

ipt

2 in

dica

tes

that

th

is i

s a

bin

ary

nu

mbe

r.)

1001

101 2

�1

�2 �

6 ��

0 �

2 �5 �

�0

�2 �

4 ��

1 �

2 �3 �

�1

�2 �

2 ��

0 �

2 �1 �

�1

�2 �

0 ��

1 �

6 �4 �

�0

�3 �2 �

�0

�1 �6 �

�1

�8 �

�1

�4 �

�0

�2 �

�1

�1 �

�64

�0

�0

�8

�4

�0

�1

�77

Fin

d t

he

dec

imal

val

ue

of e

ach

bin

ary

nu

mb

er.

1.11

112

152.

1000

0 216

3.11

0000

112

195

4.10

1110

012

185

Wri

te e

ach

dec

imal

nu

mb

er a

s a

bin

ary

nu

mb

er.

5.8

1000

6.11

1011

7.29

1110

18.

117

1110

101

9.T

he

char

t at

th

e ri

ght

show

s a

set

of d

ecim

al

code

nu

mbe

rs t

hat

is

use

d w

idel

y in

sto

rin

g le

tter

s of

th

e al

phab

et i

n a

com

pute

r's

mem

ory.

Fin

d th

e co

de n

um

bers

for

th

e le

tter

s of

you

r n

ame.

Th

en w

rite

th

e co

de f

or y

our

nam

e u

sin

g bi

nar

y n

um

bers

. An

swer

s w

ill v

ary.

Th

e A

mer

ican

Sta

nd

ard

Gu

ide

for

Info

rmat

ion

Inte

rch

ang

e (A

SC

II)

A65

N78

a97

n11

0B

66O

79b

98o

111

C67

P80

c99

p11

2D

68Q

81d

100

q11

3E

69R

82e

101

r11

4F

70S

83f

102

s11

5G

71T

84g

103

t11

6H

72U

85h

104

u11

7I

73V

86i

105

v11

8J

74W

87j

106

w11

9K

75X

88k

107

x12

0L

76Y

89l

108

y12

1M

77Z

90m

109

z12

2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-1

8-1



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Div

idin

g M

on

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-2

8-2

©G

lenc

oe/M

cGra

w-H

ill46

1G

lenc

oe A

lgeb

ra 1

Lesson 8-2

Qu

oti

ents

of

Mo

no

mia

lsT

o di

vide

tw

o po

wer

s w

ith

th

e sa

me

base

,su

btra

ct t

he

expo

nen

ts.

Qu

oti

ent

of

Po

wer

sF

or a

ll in

tege

rs m

and

nan

d an

y no

nzer

o nu

mbe

r a,

�

am�

n .

Po

wer

of

a Q

uo

tien

tF

or a

ny in

tege

r m

and

any

real

num

bers

aan

d b,

b�

0, �

�m�

.a

m� b

ma � b

am

� an

Sim

pli

fy

.Ass

um

e

nei

ther

an

or b

is e

qu

al t

o ze

ro.

��

��

Gro

up p

ower

s w

ith t

he s

ame

base

.

�(a

4 �

1 )(b

7 �

2 )Q

uotie

nt o

f P

ower

s

�a3

b5S

impl

ify.

Th

e qu

otie

nt

is a

3 b5

.

b7� b2

a4� a

a4 b7

� ab2

a4 b

7� a

b2S

imp

lify

��3 .

Ass

um

e th

at b

is n

ot e

qu

al t

o ze

ro.

��3

�P

ower

of

a Q

uotie

nt

�P

ower

of

a P

rodu

ct

�P

ower

of

a P

ower

�Q

uotie

nt o

f P

ower

s

Th

e qu

otie

nt

is

.8a

9 b9

�27

8a9 b

9�

27

8a9 b

15�27

b6

23 (a3 )

3 (b5 )

3�

�(3

)3 (b2 )

3

(2a3 b

5 )3

�(3

b2 )3

2a3 b

5�

3b2

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125

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Neg

ativ

e Ex

po

nen

tsA

ny

non

zero

nu

mbe

r ra

ised

to

the

zero

pow

er i

s 1;

for

exam

ple,

(�0.

5)0

�1.

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y n

onze

ro n

um

ber

rais

ed t

o a

neg

ativ

e po

wer

is

equ

al t

o th

e re

cipr

ocal

of

the

nu

mbe

r ra

ised

to

the

oppo

site

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er;f

or e

xam

ple,

6�3

�.T

hes

e de

fin

itio

ns

can

be

use

d to

sim

plif

y ex

pres

sion

s th

at h

ave

neg

ativ

e ex

pon

ents

.

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o E

xpo

nen

tF

or a

ny n

onze

ro n

umbe

r a,

a0

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ativ

e E

xpo

nen

t P

rop

erty

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any

non

zero

num

ber

aan

d an

y in

tege

r n,

a�

n�

and

�an

.

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e si

mpl

ifie

d fo

rm o

f an

exp

ress

ion

con

tain

ing

neg

ativ

e ex

pon

ents

mu

st c

onta

in o

nly

posi

tive

exp

onen

ts.

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pli

fy

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um

e th

at t

he

den

omin

ator

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ual

to

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ith t

he s

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pert

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ify.

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e so

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me

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omin

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al t

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Exam

ple

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8-2

8-2

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Lesson 8-2

Sim

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fy.A

ssu

me

that

no

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ator

is

equ

al t

o ze

ro.

1.61

or

62.

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r 65

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1.84

or

4096

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28.B

IOLO

GY

A l

ab t

ech

nic

ian

dra

ws

a sa

mpl

e of

blo

od.A

cu

bic

mil

lim

eter

of

the

bloo

dco

nta

ins

223

wh

ite

bloo

d ce

lls

and

225

red

bloo

d ce

lls.

Wh

at i

s th

e ra

tio

of w

hit

e bl

ood

cell

s to

red

blo

od c

ells

?

29. C

OU

NTI

NG

Th

e n

um

ber

of t

hre

e-le

tter

“w

ords

”th

at c

an b

e fo

rmed

wit

h t

he

En

glis

hal

phab

et i

s 26

3 .T

he

nu

mbe

r of

fiv

e-le

tter

“w

ords

”th

at c

an b

e fo

rmed

is

265 .

How

man

yti

mes

mor

e fi

ve-l

ette

r “w

ords

”ca

n b

e fo

rmed

th

an t

hre

e-le

tter

“w

ords

”?67

6

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4

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8-2

8-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csD

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no

mia

ls

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8-2

8-2

©G

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5G

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Lesson 8-2

Pre-

Act

ivit

yH

ow c

an y

ou c

omp

are

pH

lev

els?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-2

at

the

top

of p

age

417

in y

our

text

book

.

•In

th

e fo

rmu

la c

��

�pH,i

den

tify

th

e ba

se a

nd

the

expo

nen

t.

bas

e �

,exp

on

ent

�p

H

•H

ow d

o yo

u t

hin

k c

wil

l ch

ange

as

the

expo

nen

t in

crea

ses?

cw

ill d

ecre

ase.

Rea

din

g t

he

Less

on

1.E

xpla

in w

hat

th

e st

atem

ent

�am

�n

mea

ns.

To d

ivid

e tw

o p

ow

ers

that

hav

e th

e sa

me

bas

e,su

btr

act

the

exp

on

ents

.

2.To

fin

d c

in t

he f

orm

ula

c�

��pH

,you

can

fin

d th

e po

wer

of

the

num

erat

or,t

he p

ower

of

the

den

omin

ator

,an

d di

vide

.Th

is i

s an

exa

mpl

e of

wh

at p

rope

rty?

Po

wer

of

a Q

uo

tien

t P

rop

erty

3.U

se t

he

Qu

otie

nt

of P

ower

s P

rope

rty

to e

xpla

in w

hy

30�

1.S

amp

le a

nsw

er:

�1.

Th

e Q

uo

tien

t o

f P

ow

ers

Pro

per

ty s

ays

that

wh

en y

ou

div

ide

two

po

wer

s th

at h

ave

the

sam

e b

ase,

you

su

btr

act

the

exp

on

ents

.

So

�

30.

4.C

onsi

der

the

expr

essi

on 4

�3 .

a.E

xpla

in w

hy

the

expr

essi

on 4

�3

is n

ot s

impl

ifie

d.A

n e

xpre

ssio

n in

volv

ing

exp

on

ents

is n

ot

con

sid

ered

sim

plif

ied

if t

he

exp

ress

ion

co

nta

ins

neg

ativ

e ex

po

nen

ts.

b.

Def

ine

the

term

rec

ipro

cal.

Th

e re

cip

roca

l of

a n

um

ber

is 1

div

ided

by

the

nu

mb

er.

c.4�

3is

th

e re

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ocal

of

wh

at p

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of

4?43

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at i

s th

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mpl

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d fo

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r

Hel

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g Y

ou

Rem

emb

er

5.D

escr

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how

you

wou

ld h

elp

a fr

ien

d w

ho

nee

ds t

o si

mpl

ify

the

expr

essi

on

.

Div

ide

the

con

stan

ts a

nd

gro

up

po

wer

s w

ith

th

e sa

me

bas

e to

get

��

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th

e Q

uo

tien

t o

f P

ow

ers

Pro

per

ty t

o g

et (

2)(x

2�5 )

or

(2)(

x�3).

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imp

lify

(2)(

x�3 )

,use

th

e N

egat

ive

Exp

on

ent

Pro

per

ty t

o g

et

(2) �

�,or

.2 � x3

1 � x3x2� x5

4 � 2

4x2

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1 � 641 � 43

34� 34

34� 34

1 � 10am� an

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Pat

tern

s w

ith

Pow

ers

Use

you

r ca

lcu

lato

r,if

nec

essa

ry,t

o co

mpl

ete

each

pat

tern

.

a.21

0�

b.

510

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410

�

29�

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52�

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Stu

dy

the

pat

tern

s fo

r a,

b,a

nd

c a

bov

e.T

hen

an

swer

th

e q

ues

tion

s.

1.D

escr

ibe

the

patt

ern

of

the

expo

nen

ts f

rom

th

e to

p of

eac

h c

olu

mn

to

the

bott

om.

Th

e ex

po

nen

ts d

ecre

ase

by o

ne

fro

m e

ach

ro

w t

o t

he

on

e b

elo

w.

2.D

escr

ibe

the

patt

ern

of

the

pow

ers

from

th

e to

p of

th

e co

lum

n t

o th

e bo

ttom

.To

get

each

po

wer

,div

ide

the

po

wer

on

th

e ro

w a

bov

e by

th

e b

ase

(2,5

,or

4).

3.W

hat

wou

ld y

ou e

xpec

t th

e fo

llow

ing

pow

ers

to b

e?20

150

140

1

4.R

efer

to

Exe

rcis

e 3.

Wri

te a

ru

le.T

est

it o

n p

atte

rns

that

you

obt

ain

usi

ng

22,2

5,an

d 24

as b

ases

.A

ny n

on

zero

nu

mb

er t

o t

he

zero

po

wer

eq

ual

s o

ne.

Stu

dy

the

pat

tern

bel

ow.T

hen

an

swer

th

e q

ues

tion

s.

03�

002

�0

01�

000

�0�

1do

es n

ot e

xist

.0�

2do

es n

ot e

xist

.0�

3do

es n

ot e

xist

.

5.W

hy

do 0

�1 ,

0�2 ,

and

0�3

not

exi

st?

Neg

ativ

e ex

po

nen

ts a

re n

ot

def

ined

un

less

th

e b

ase

is n

on

zero

.

6.B

ased

upo

n t

he

patt

ern

,can

you

det

erm

ine

wh

eth

er 0

0ex

ists

? N

o,s

ince

th

e p

atte

rn 0

n�

0 b

reak

s d

ow

n f

or

n�

1.

7.T

he

sym

bol

00is

cal

led

an i

nd

eter

min

ate,

wh

ich

mea

ns

that

it

has

no

un

iqu

e va

lue.

Thu

s it

doe

s no

t ex

ist

as a

uni

que

real

num

ber.

Why

do

you

thin

k th

at 0

0ca

nnot

equ

al 1

?

An

swer

s w

ill v

ary.

On

e an

swer

is t

hat

if 0

0�

1,th

en 1

��

��

�0 ,

wh

ich

is a

fal

se r

esu

lt,s

ince

div

isio

n b

y ze

ro is

no

t al

low

ed.T

hu

s,00

can

no

t eq

ual

1.

1 � 010� 00

1 � 1

?

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____

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ER

IOD

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8-2

8-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Sci

enti

fic

No

tati

on

NA

ME

____

____

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6-3

8-3

©G

lenc

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cGra

w-H

ill46

7G

lenc

oe A

lgeb

ra 1

Lesson 8-3

Scie

nti

fic

No

tati

on

Kee

pin

g tr

ack

of p

lace

val

ue

in v

ery

larg

e or

ver

y sm

all

nu

mbe

rsw

ritt

en i

n s

tan

dard

for

m m

ay b

e di

ffic

ult

.It

is m

ore

effi

cien

t to

wri

te s

uch

nu

mbe

rs i

nsc

ien

tifi

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8-3



Readin

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e in

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son

8-3

at

the

top

of p

age

425

in y

our

text

book

.

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he

tabl

e,ea

ch m

ass

is w

ritt

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s th

e pr

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nu

mbe

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10.

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th

e fi

rst

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pro

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ese

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ey a

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1 a

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m s

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not

atio

n t

o sc

ien

tifi

c n

otat

ion

.

a.T

o ex

pres

s 0.

0007

865

in s

cien

tifi

c n

otat

ion

,mov

e th

e de

cim

al p

oin

t pl

aces

to t

he

righ

t an

d w

rite

.

b.

To

expr

ess

54,0

00,0

00,0

00 i

n s

cien

tifi

c n

otat

ion

,mov

e th

e de

cim

al p

oin

t

plac

es t

o th

e le

ft a

nd

wri

te

.

4.W

rite

pos

itiv

eor

neg

ativ

eto

com

plet

e ea

ch s

ente

nce

.

a.po

wer

s of

10

are

use

d to

exp

ress

ver

y la

rge

nu

mbe

rs i

n

scie

nti

fic

not

atio

n.

b.

pow

ers

of 1

0 ar

e u

sed

to e

xpre

ss v

ery

smal

l n

um

bers

in

scie

nti

fic

not

atio

n.

Hel

pin

g Y

ou

Rem

emb

er

5.D

escr

ibe

the

met

hod

you

wou

ld u

se t

o es

tim

ate

how

man

y ti

mes

gre

ater

th

e m

ass

ofS

atu

rn i

s th

an t

he

mas

s of

Plu

to.

Div

ide

5.69

�10

26by

1.2

7 �

1022

.Sin

ce 5

.69

�1.

27 �

4.48

an

d

1026

�10

22is

104

,th

e m

ass

of

Mar

s is

ab

ou

t 4.

48 �

104

tim

es t

he

mas

s o

f P

luto

.

Neg

ativ

e

Po

siti

ve

5.4

�10

10

107.

865

�10

�4

4

left

3

rig

ht

6

©G

lenc

oe/M

cGra

w-H

ill47

2G

lenc

oe A

lgeb

ra 1

Co

nver

tin

g M

etri

c U

nit

sS

cien

tifi

c n

otat

ion

is

con

ven

ien

t to

use

for

un

it c

onve

rsio

ns

in t

he

met

ric

syst

em.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-3

8-3

How

man

y k

ilom

eter

sar

e th

ere

in 4

,300

,000

met

ers?

Div

ide

the

mea

sure

by

the

nu

mbe

r of

met

ers

(100

0) i

n o

ne

kilo

met

er.E

xpre

ssbo

th n

um

bers

in

sci

enti

fic

not

atio

n.

�4.

3 �

103

Th

e an

swer

is

4.3

3 10

3 km

.

4.3

�10

6�

�1

�10

3

Con

vert

370

0 gr

ams

into

mil

ligr

ams.

Mu

ltip

ly b

y th

e n

um

ber

of m

illi

gram

s(1

000)

in

1 g

ram

.

(3.7

�10

3 )(1

�10

3 ) �

3.7

�10

6

Th

ere

are

3.7

�10

6m

g in

370

0 g.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Com

ple

te t

he

foll

owin

g.E

xpre

ss e

ach

an

swer

in

sci

enti

fic

not

atio

n.

1.25

0,00

0 m

�km

2.37

5 km

�m

3.24

7 m

�cm

4.50

00 m

�m

m

5.0.

0004

km

�m

6.0.

01 m

m �

m

7.60

00 m

�m

m8.

340

cm �

km

9.52

,000

mg

�g

10.4

20 k

L �

L

Sol

ve.

11.T

he

plan

et M

ars

has

a d

iam

eter

of

6.76

�10

3km

.Wh

at i

s th

edi

amet

er o

f M

ars

in m

eter

s? E

xpre

ss t

he

answ

er i

n b

oth

sci

enti

fic

and

deci

mal

not

atio

n.

6,76

0,00

0 m

;6.

76 �

106

m

12.T

he

dist

ance

fro

m e

arth

to

the

sun

is

149,

590,

000

km.L

igh

ttr

avel

s 3.

0 �

108

met

ers

per

seco

nd.

How

lon

g do

es i

t ta

ke l

igh

tfr

om t

he

sun

to

reac

h t

he

eart

h i

n m

inu

tes?

Rou

nd

to t

he

nea

rest

hu

ndr

edth

.8.

31 m

in

13.A

lig

ht-

year

is

the

dist

ance

th

at l

igh

t tr

avel

s in

on

e ye

ar.(

See

Exe

rcis

e 12

.) H

ow f

ar i

s a

ligh

t ye

ar i

n k

ilom

eter

s? E

xpre

ss y

our

answ

er i

n s

cien

tifi

c n

otat

ion

.Rou

nd

to t

he

nea

rest

hu

ndr

edth

.9.

46 �

1012

km

4.2

�10

55.

2 �

101

3.4

�10

�3

6 �

106

1 �

10�

54

�10

�1

5 �

106

2.47

�10

4

3.75

�10

52.

5 �

102



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

6-4

8-4

©G

lenc

oe/M

cGra

w-H

ill47

3G

lenc

oe A

lgeb

ra 1

Lesson 8-4

Deg

ree

of

a Po

lyn

om

ial

A p

olyn

omia

lis

a m

onom

ial

or a

su

m o

f m

onom

ials

.Ab

inom

ial

is t

he

sum

of

two

mon

omia

ls,a

nd

a tr

inom

ial

is t

he

sum

of

thre

e m

onom

ials

.P

olyn

omia

ls w

ith

mor

e th

an t

hre

e te

rms

hav

e n

o sp

ecia

l n

ame.

Th

e d

egre

eof

a m

onom

ial

is t

he

sum

of

the

expo

nen

ts o

f al

l it

s va

riab

les.

Th

e d

egre

e of

th

e p

olyn

omia

lis

th

e sa

me

as t

he

degr

ee o

f th

e m

onom

ial

term

wit

h t

he

hig

hes

t de

gree

.

Sta

te w

het

her

eac

h e

xpre

ssio

n i

s a

pol

ynom

ial.

If t

he

exp

ress

ion

is

a p

olyn

omia

l,id

enti

fy i

t as

a m

onom

ial,

bin

omia

l,or

tri

nom

ial.

Th

en g

ive

the

deg

ree

of t

he

pol

ynom

ial.

Exp

ress

ion

Po

lyn

om

ial?

Mo

no

mia

l,B

ino

mia

l,D

egre

e o

f th

e o

r Tri

no

mia

l?P

oly

no

mia

l

3x�

7xyz

Yes.

3x

�7x

yz�

3x�

(�7x

yz),

bino

mia

l3

whi

ch is

the

sum

of

two

mon

omia

ls

�25

Yes.

�25

is a

rea

l num

ber.

mon

omia

l0

7n3

�3n

�4

No.

3n�

4�

, w

hich

is n

ot

none

of

thes

e—

a m

onom

ial

Yes.

The

exp

ress

ion

sim

plifi

es t

o9 x

3�

4x�

x�

4 �

2x9x

3�

7x�

4, w

hich

is t

he s

um

trin

omia

l3

of t

hree

mon

omia

ls

Sta

te w

het

her

eac

h e

xpre

ssio

n i

s a

pol

ynom

ial.

If t

he

exp

ress

ion

is

a p

olyn

omia

l,id

enti

fy i

t as

a m

onom

ial,

bin

omia

l,or

tri

nom

ial.

1.36

yes;

mo

no

mia

l2.

�5

no

3.7x

�x

�5

yes;

bin

om

ial

4.8g

2 h�

7gh

�2

yes;

trin

om

ial

5.�

5y�

8n

o6.

6x�

x2ye

s;b

ino

mia

l

Fin

d t

he

deg

ree

of e

ach

pol

ynom

ial.

7.4x

2 y3 z

68.

�2a

bc3

9.15

m1

10.s

�5t

111

.22

012

.18x

2�

4yz

�10

y2

13.x

4�

6x2

�2x

3�

104

14.2

x3y2

�4x

y35

15.�

2r8 s

4�

7r2 s

�4r

7 s6

13

16.9

x2�

yz8

917

.8b

�bc

56

18.4

x4y

�8z

x2�

2x5

5

19.4

x2�

12

20.9

abc

�bc

�d

55

21.h

3 m�

6h4 m

2�

76

1� 4y

2

3 � q2

3 � n4

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill47

4G

lenc

oe A

lgeb

ra 1

Wri

te P

oly

no

mia

ls in

Ord

erT

he

term

s of

a p

olyn

omia

l ar

e u

sual

ly a

rran

ged

so t

hat

th

e po

wer

s of

on

e va

riab

le a

re i

n a

scen

din

g(i

ncr

easi

ng)

ord

er o

r d

esce

nd

ing

(dec

reas

ing)

ord

er.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

Arr

ange

th

e te

rms

ofea

ch p

olyn

omia

l so

th

at t

he

pow

ers

ofx

are

in a

scen

din

g or

der

.

a.x4

�x2

5x

3

�x2

�5x

3�

x4

b.

8x3 y

�y2

6x

2 y

xy2

�y2

�xy

2�

6x2 y

�8x

3 y

Arr

ange

th

e te

rms

ofea

ch p

olyn

omia

l so

th

at t

he

pow

ers

ofx

are

in d

esce

nd

ing

ord

er.

a.x4

4x

5�

x2

4x5

�x4

�x2

b.

�6x

y

y3�

x2y2

x4

y2

x4y2

� x

2 y2

�6x

y�

y3

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Arr

ange

th

e te

rms

of e

ach

pol

ynom

ial

so t

hat

th

e p

ower

s of

xar

e in

as

cen

din

g or

der

.

1.5x

�x2

�6

2.6x

�9

�4x

23.

4xy

�2y

�6x

2

6

5x

x2

9

6x�

4x2

2y

4xy

6x

2

4.6y

2 x�

6x2 y

�2

5.x4

�x3

�x2

6.2x

3�

x�

3x7

2

6y2 x

�6x

2 yx

2

x3

x4

�x

2x

3 3x

7

7.�

5cx

�10

c2x3

�15

cx2

8.�

4nx

�5n

3 x3 �

59.

4xy

�2y

�5x

2

�5c

x

15cx

2

10c

2 x3

5 �

4nx

�5n

3 x3

2y

4xy

5x

2

Arr

ange

th

e te

rms

of e

ach

pol

ynom

ial

so t

hat

th

e p

ower

s of

xar

e in

d

esce

nd

ing

ord

er.

10.2

x�

x2�

511

.20x

�10

x2�

5x3

12.x

2�

4yx�

10x5

x2

2x�

55x

3�

10x2

20

x�

10x5

x2

4y

x

13.9

bx�

3bx2

�6x

314

.x3

�x5

�x2

15.a

x2�

8a2 x

5�

4

�6x

3

3bx

2

9bx

x5

x

3�

x2

8a2 x

5

ax2

�4

16.3

x3y

�4x

y2�

x4y2

�y5

17.x

4�

4x3

�7x

5�

1

�x

4 y2

3x

3 y�

4xy

2

y5

�7x

5

x4

4x

3

1

18.�

3x6

�x5

�2x

819

.�15

cx2

�8c

2 x5

�cx

2x8

�3x

6�

x58c

2 x5

�15

cx2

cx

20.2

4x2 y

�12

x3y2

�6x

421

.�15

x3�

10x4

y2�

7xy2

6x4

�12

x3 y

2

24x

2 y10

x4 y

2�

15x

3

7xy

2



Skil

ls P

ract

ice

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill47

5G

lenc

oe A

lgeb

ra 1

Lesson 8-4

Sta

te w

het

her

eac

h e

xpre

ssio

n i

s a

pol

ynom

ial.

If t

he

exp

ress

ion

is

a p

olyn

omia

l,id

enti

fy i

t as

a m

onom

ial,

a bi

nom

ial,

or a

tri

nom

ial.

1.5m

n�

n2

2.4b

y�

2b�

by3.

�32

yes;

bin

om

ial

yes;

bin

om

ial

yes;

mo

no

mia

l

4.5.

5x2

�3x

�4

6.2c

2�

8c�

9 �

3

yes;

mo

no

mia

ln

oye

s;tr

ino

mia

l

GEO

MET

RYW

rite

a p

olyn

omia

l to

rep

rese

nt

the

area

of

each

sh

aded

reg

ion

.

7.ab

�xy

8.4r

2�

�r2

Fin

d t

he

deg

ree

of e

ach

pol

ynom

ial.

9.12

010

.3r4

411

.b�

61

12.4

a3�

2a3

13.5

abc

�2b

2�

13

14.8

x5y4

�2x

89

Arr

ange

th

e te

rms

of e

ach

pol

ynom

ial

so t

hat

th

e p

ower

s of

xar

e in

asc

end

ing

ord

er.

15.3

x�

1 �

2x2

1

3x

2x2

16.5

x�

6 �

3x2

�6

5x

3x

2

17.9

x2�

2 �

x3�

x2

x

9x

2

x3

18.�

3 �

3x3

�x2

�4x

�3

4x

�x

2

3x3

19.7

r5x

�21

r4�

r2x2

�15

x320

.3a2

x4�

14a2

�10

x3�

ax2

21r4

7r

5 x�

r2x

2�

15x3

14a2

ax

2�

10x3

3a

2 x4

Arr

ange

th

e te

rms

of e

ach

pol

ynom

ial

so t

hat

th

e p

ower

s of

xar

e in

des

cen

din

gor

der

.

21.x

2�

3x3

�27

�x

22.2

5 �

x3�

x

3x3

x2

�x

27

�x3

x

25

23.x

�3x

2�

4 �

5x3

24.x

2�

64 �

x�

7x3

5x3

�3x

2

x

47x

3

x2�

x

64

25.2

cx�

32 �

c3x2

�6x

326

.13

�x3

y3�

x2y2

�x

6x3

�c

3 x2

2c

x

32�

x3y

3

x2y

2

x

13

r

ax

by

3x � 7

©G

lenc

oe/M

cGra

w-H

ill47

6G

lenc

oe A

lgeb

ra 1

Sta

te w

het

her

eac

h e

xpre

ssio

n i

s a

pol

ynom

ial.

If t

he

exp

ress

ion

is

a p

olyn

omia

l,id

enti

fy i

t as

a m

onom

ial,

a bi

nom

ial,

or a

tri

nom

ial.

1.7a

2 b�

3b2

�a2

b2.

y3�

y2�

93.

6g2 h

3 k

yes;

bin

om

ial

yes;

trin

om

ial

yes;

mo

no

mia

l

GEO

MET

RYW

rite

a p

olyn

omia

l to

rep

rese

nt

the

area

of

each

sh

aded

reg

ion

.

4.ab

�b

25.

d2

��

d2

Fin

d t

he

deg

ree

of e

ach

pol

ynom

ial.

6.x

�3x

4�

21x2

�x3

47.

3g2 h

3�

g3h

5

8.�

2x2 y

�3x

y3�

x24

9.5n

3 m�

2m3

�n

2 m4

�n

26

10.a

3 b2 c

�2a

5 c�

b3c2

611

.10s

2 t2

�4s

t2�

5s3 t

25

Arr

ange

th

e te

rms

of e

ach

pol

ynom

ial

so t

hat

th

e p

ower

s of

xar

e in

asc

end

ing

ord

er.

12.8

x2�

15 �

5x5

13.1

0bx

�7b

2�

x4�

4b2 x

3

�15

8x

2

5x5

�7b

2

10b

x

4b2 x

3

x4

14.�

3x3 y

�8y

2�

xy4

15.7

ax�

12 �

3ax3

�a2

x2

8y2

xy

4�

3x3 y

�12

7a

x

a2x2

3a

x3

Arr

ange

th

e te

rms

of e

ach

pol

ynom

ial

so t

hat

th

e p

ower

s of

xar

e in

des

cen

din

gor

der

.

16.1

3x2

�5

�6x

3�

x17

.4x

�2x

5�

6x3

�2

6x3

13

x2�

x�

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22.M

ON

EYW

rite

a p

olyn

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l to

rep

rese

nt

the

valu

e of

tte

n-d

olla

r bi

lls,

ffi

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doll

arbi

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and

hon

e-h

un

dred

-dol

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bill

s.10

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10

0h

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RA

VIT

YT

he

hei

ght

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e gr

oun

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a b

all

thro

wn

up

wit

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vel

ocit

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96

feet

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from

a h

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6 f

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�96

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e ti

me

in s

econ

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to t

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ow h

igh

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the

ball

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sec

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xpla

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e h

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bal

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g t

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fee

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Po

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NA

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____

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____

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8-4

8-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csP

oly

no

mia

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NA

ME

____

____

____

____

____

____

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____

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AT

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IOD

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8-4

8-4

©G

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w-H

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7G

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Lesson 8-4

Pre-

Act

ivit

yH

ow a

re p

olyn

omia

ls u

sefu

l in

mod

elin

g d

ata?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-4

at

the

top

of p

age

432

in y

our

text

book

.

•H

ow m

any

term

s do

es t

4�

9t3

�26

t�

18t

�76

hav

e?

five

•W

hat

cou

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ou c

all

a po

lyn

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ith

just

on

e te

rm?

a m

on

om

ial

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din

g t

he

Less

on

1.W

hat

is

the

mea

nin

g of

th

e pr

efix

es m

ono-

,bi-

,an

d tr

i-?

Mo

no

- m

ean

s o

ne,

bi-

mea

ns

two

,an

d t

ri-

mea

ns

thre

e.

2.W

rite

exa

mpl

es o

f w

ords

th

at b

egin

wit

h t

he

pref

ixes

mon

o-,b

i-,a

nd

tri-

.

Sam

ple

an

swer

:m

on

ocy

cle

(on

e w

hee

l),b

icyc

le (

two

wh

eels

),tr

icyc

le(t

hre

e w

hee

ls)

3.C

ompl

ete

the

tabl

e. mo

no

mia

lb

ino

mia

ltr

ino

mia

lp

oly

no

mia

l wit

h m

ore

th

an t

hre

e te

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hat

is

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degr

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3xy2

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hat

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the

degr

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f th

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4�

2x3 y

3�

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how

you

fou

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you

r an

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6;S

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0

4 �

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x4

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deg

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0

3

3 �

6,2x

3 y3

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deg

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6;y

2h

as d

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and

14

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he

hig

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term

s is

6.

Hel

pin

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6.U

se a

dic

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to f

ind

the

mea

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th

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rms

asce

nd

ing

and

des

cen

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rite

th

eir

mea

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gs a

nd

then

des

crib

e a

situ

atio

n i

n y

our

ever

yday

lif

e th

at r

elat

es t

o th

em.

asce

nd

ing

:g

oin

g,g

row

ing

,or

mov

ing

up

war

d;

des

cen

din

g:

mov

ing

fro

ma

hig

her

to

a lo

wer

pla

ce;

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ple

an

swer

:cl

imb

ing

sta

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hik

ing

©G

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Po

lyn

om

ial F

un

ctio

ns

Su

ppos

e a

lin

ear

equ

atio

n s

uch

as

23x

�y

�4

is

sol

ved

for

y.T

hen

an

equ

ival

ent

equ

atio

n,

y�

3x�

4,is

fou

nd.

Exp

ress

ed i

n t

his

way

,yis

a

fun

ctio

n o

f x,

or f

(x)

�3x

�4.

Not

ice

that

th

e ri

ght

side

of

the

equ

atio

n i

s a

bin

omia

l of

de

gree

1.

Hig

her

-deg

ree

poly

nom

ials

in

xm

ay a

lso

form

fu

nct

ion

s.A

n e

xam

ple

is f

(x)

�x3

�1,

wh

ich

is

a p

olyn

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l fu

nct

ion

of

degr

ee 3

.You

can

gr

aph

th

is f

un

ctio

n u

sin

g a

tabl

e of

ord

ered

pa

irs,

as s

how

n a

t th

e ri

ght.

For

eac

h o

f th

e fo

llow

ing

pol

ynom

ial

fun

ctio

ns,

mak

e a

tab

leof

val

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for

xan

d y

�f(

x).T

hen

dra

w t

he

grap

h o

n t

he

grid

.

1.f(

x) �

1 �

x22.

f(x)

�x2

�5

3.f(

x) �

x2�

4x�

14.

f(x)

�x3

x

y

Ox

y

O

x

y

Ox

y

O

x

y

O

xy

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1 2��

2 �3 8�

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01

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1 �1 2�

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rich

men

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NA

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____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

____

_

8-4

8-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Add

ing

an

d S

ub

trac

tin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

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_

6-5

8-5

©G

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w-H

ill47

9G

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Lesson 8-5

Ad

d P

oly

no

mia

lsTo

add

pol

ynom

ials

,you

can

gro

up li

ke t

erm

s ho

rizo

ntal

ly o

r w

rite

them

in

col

um

n f

orm

,ali

gnin

g li

ke t

erm

s ve

rtic

ally

.Lik

e te

rms

are

mon

omia

l te

rms

that

are

eith

er i

den

tica

l or

dif

fer

only

in

th

eir

coef

fici

ents

,su

ch a

s 3p

and

�5p

or 2

x2y

and

8x2 y

.

Fin

d

(2x2

x

�8)

(3

x�

4x2

2)

.

Hor

izon

tal

Met

hod

Gro

up

like

ter

ms.

(2x2

�x

�8)

�(3

x�

4x2

�2)

�[(

2x2

�(�

4x2 )

] �

(x�

3x)

�[(

�8)

�2)

]�

�2x

2�

4x�

6.

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e su

m i

s �

2x2

�4x

�6.

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d (

3x2

5x

y)

(xy

2x

2 ).

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tica

l M

eth

odA

lign

lik

e te

rms

in c

olu

mn

s an

d ad

d.

3x2

�5x

y(�

)2x

2�

xyP

ut t

he t

erm

s in

des

cend

ing

orde

r.

5x2

�6x

y

Th

e su

m i

s 5x

2�

6xy.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d e

ach

su

m.

1.(4

a�

5) �

(3a

�6)

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x�

9) �

(4x2

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7a

14x

2

6x

2

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xy�

2y�

6x)

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xy�

x)4.

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�y2

) �

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�y2

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10xy

5x

2y

2y2

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p2�

2p�

3) �

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�7p

�7)

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x2�

5xy

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10

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xy�

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2a�

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2p�

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2x2

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2) �

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11

15.(

3z2

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) �

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�2z

) �

(z�

4)16

.(8x

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x2�

x�

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x2

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Sub

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om

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You

can

su

btra

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pol

ynom

ial

by a

ddin

g it

s ad

diti

ve i

nve

rse.

To

fin

d th

e ad

diti

ve i

nve

rse

of a

pol

ynom

ial,

repl

ace

each

ter

m w

ith

its

add

itiv

e in

vers

e or

opp

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e.

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d (

3x2

2x

�6)

�(2

x

x2

3).

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dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Add

ing

an

d S

ub

trac

tin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

Exam

ple

Exam

ple

Hor

izon

tal

Met

hod

Use

add

itiv

e in

vers

es t

o re

wri

te a

s ad

diti

on.

Th

en g

rou

p li

ke t

erm

s.

(3x2

�2x

�6)

�(2

x�

x2�

3)�

(3x2

�2x

�6)

�[(

�2x

)�(�

x2)

�(�

3)]

�[3

x2�

(�x2

)] �

[2x

�(�

2x)]

�[�

6 �

(�3)

]�

2x2

�(�

9)�

2x2

�9

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e di

ffer

ence

is

2x2

�9.

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tica

l M

eth

od

Ali

gn l

ike

term

s in

col

um

ns

and

subt

ract

by

addi

ng

the

addi

tive

in

vers

e.

3x2

�2x

�6

(�)

x2�

2x�

3

3x2

�2x

�6

(�)

�x2

�2x

�3

2x2

�9

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e di

ffer

ence

is

2x2

�9.

Exer

cises

Exer

cises

Fin

d e

ach

dif

fere

nce

.

1.(3

a�

5) �

(5a

�1)

2.(9

x�

2) �

(�3x

2�

5)

�2a

�6

3x2

9x

7

3.(9

xy�

y�

2x)

�(6

xy�

2x)

4.(x

2�

y2)

�(�

x2�

y2)

3xy

y

2x2

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p2�

4p�

5) �

(2p2

�5p

�1)

6.(6

x2�

5xy

�2y

2 ) �

(�xy

�2x

2�

4y2 )

4p2

9p

4

8x2

6x

y

2y2

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p�

5q)

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6p2

�6q

�3)

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x2�

4x�

3) �

(�2x

�x2

�5)

6p2

8p

�11

q

39x

2�

2x�

8

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x2�

2x)

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x2�

5x�

1)10

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2�

6xy

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2 ) �

(�x2

�2x

y�

5y2 )

�7x

1

5x2

4xy

7y

2

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2h�

6j�

2k)

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7h�

5j�

4k)

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9xy2

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8xy2

)

9h�

j2k

17xy

2

7xy

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2a�

8b)

�(�

3a�

5b)

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2x2

�8)

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2x2

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4x2

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6z2

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z2�

z)16

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2x�

4)

2z2

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13x2

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s

Skil

ls P

ract

ice

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ing

an

d S

ub

trac

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g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

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Lesson 8-5

Fin

d e

ach

su

m o

r d

iffe

ren

ce.

1.(2

x�

3y)

�(4

x�

9y) 6x

12

y2.

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) 14

s

9t

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a�

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a�

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5b

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n

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2�

m)

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m�

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2

m6.

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) �

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) �

x2�

8x

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2�

d�

5) �

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e�

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f�

g�

2) �

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k2�

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g

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k2

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9

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x3�

x�

1) �

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b2�

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x2�

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6x2

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x

3

©G

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ra 1

Fin

d e

ach

su

m o

r d

iffe

ren

ce.

1.(4

y�

5) �

(�7y

�1)

2.(�

x2�

3x)

�(5

x�

2x2 )

�3y

4

�3x

2�

2x

3.(4

k2�

8k�

2) �

(2k

�3)

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m2

�6m

) �

(m2

�5m

�7)

4k2

6k

�1

3m2

m

7

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w2

�3w

�1)

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w�

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2 ) �

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w

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3

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�p

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2�

3p�

1)�

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13

a�

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9x�

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5

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5) �

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2 �

b3)

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n�

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1

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b�

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9d2

d

9x2

�x

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3h2

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�1)

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h�

8h2

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) �

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2

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25m

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x2�

y2�

6) �

(5x2

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2�

7 �

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23.(

k3�

2k2

�4k

�6)

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4k�

k2�

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.(9j

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j�

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3�

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8k

9

6j2

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6y�

3z)

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x�

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8y)

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�z)

7x�

5y

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26.(

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�7f

�3)

�(5

f2�

1 �

2f)

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3 �

f)�

f2�

10f

1

27.B

USI

NES

ST

he

poly

nom

ial

s3�

70s2

�15

00s

�10

,800

mod

els

the

prof

it a

com

pan

ym

akes

on

sel

lin

g an

ite

m a

t a

pric

e s.

A s

econ

d it

em s

old

at t

he

sam

e pr

ice

brin

gs i

n a

prof

it o

f s3

�30

s2�

450s

�50

00.W

rite

a p

olyn

omia

l th

at e

xpre

sses

th

e to

tal

prof

itfr

om t

he

sale

of

both

ite

ms.

2s3

�10

0s2

19

50s

�15

,800

28.G

EOM

ETRY

Th

e m

easu

res

of t

wo

side

s of

a t

rian

gle

are

give

n.

If P

is t

he

peri

met

er,a

nd

P�

10x

�5y

,fin

d th

e m

easu

re o

f th

e th

ird

side

.2x

2y

3x

4y

5x �

y

Pra

ctic

e (

Ave

rag

e)

Add

ing

an

d S

ub

trac

tin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5



Readin

g t

o L

earn

Math

em

ati

csA

ddin

g a

nd

Su

btr

acti

ng

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill48

3G

lenc

oe A

lgeb

ra 1

Lesson 8-5

Pre-

Act

ivit

yH

ow c

an a

dd

ing

pol

ynom

ials

hel

p y

ou m

odel

sal

es?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-5

at

the

top

of p

age

439

in y

our

text

book

.

Wh

at o

pera

tion

wou

ld y

ou u

se t

o fi

nd

how

mu

ch m

ore

the

trad

itio

nal

toy

sale

s R

wer

e th

an t

he

vide

o ga

mes

sal

es V

?

sub

trac

tio

n

Rea

din

g t

he

Less

on

1.U

se t

he

exam

ple

(�3x

3�

4x2

�5x

�1)

�(�

5x3

�2x

2�

2x�

7).

a.S

how

wh

at i

s m

ean

t by

gro

upi

ng

like

ter

ms

hor

izon

tall

y.

[�3x

3

(�5x

3 )]

[(

4x2

(�

2x2 )

]

(5x

2x

)

[1

(�7)

]

b.

Sh

ow w

hat

is

mea

nt

by a

lign

ing

like

ter

ms

vert

ical

ly.

�3x

3

4x2

5x

1

()

�5x

3�

2x2

2x

�7

c.C

hoo

se o

ne

met

hod

,th

en a

dd t

he

poly

nom

ials

.

�8x

3

2x2

7x

�6

2.H

ow i

s su

btra

ctin

g a

poly

nom

ial

like

su

btra

ctin

g a

rati

onal

nu

mbe

r?

You

su

btr

act

by a

dd

ing

th

e ad

dit

ive

inve

rse.

3.A

n a

lgeb

ra s

tude

nt

got

the

foll

owin

g ex

erci

se w

ron

g on

his

hom

ewor

k.W

hat

was

h

is e

rror

?

(3x5

�3x

4�

2x3

�4x

2�

5) �

(2x5

�x3

�2x

2�

4)�

[3x5

�(�

2x5 )

] �

(�3x

4 ) �

[2x3

�(�

x3)]

�[�

4x2

�(�

2x2 )

] �

(5 �

4)�

x5�

3x4

�x3

�6x

2�

9

He

did

no

t ad

d t

he

add

itiv

e in

vers

e o

f �

x3 .

Hel

pin

g Y

ou

Rem

emb

er

4.H

ow i

s ad

din

g an

d su

btra

ctin

g po

lyn

omia

ls v

erti

call

y li

ke a

ddin

g an

d su

btra

ctin

gde

cim

als

vert

ical

ly?

Alig

nin

g li

ke t

erm

s w

hen

ad

din

g o

r su

btr

acti

ng

po

lyn

om

ials

is li

ke u

sin

gp

lace

val

ue

to a

lign

dig

its

wh

en y

ou

ad

d o

r su

btr

act

dec

imal

s.

©G

lenc

oe/M

cGra

w-H

ill48

4G

lenc

oe A

lgeb

ra 1

Cir

cula

r A

reas

an

d V

olu

mes

Are

a of

Cir

cle

Vol

um

e of

Cyl

ind

erV

olu

me

of C

one

A�

�r2

V�

�r2

hV

��

r2h

Wri

te a

n a

lgeb

raic

exp

ress

ion

for

eac

h s

had

ed a

rea.

(Rec

all

that

th

e d

iam

eter

of

a ci

rcle

is

twic

e it

s ra

diu

s.)

1.2.

3.

�x2

��

�2�

�x2

(y2

2x

y)�

x2

Wri

te a

n a

lgeb

raic

exp

ress

ion

for

th

e to

tal

volu

me

of e

ach

fig

ure

.

4.5.

5�

x3

[13x

3

(4a

9b

)x2 ]

Eac

h f

igu

re h

as a

cyl

ind

rica

l h

ole

wit

h a

rad

ius

of 2

in

ches

an

d

a h

eigh

t of

5 i

nch

es.F

ind

eac

h v

olu

me.

6.7.

x3

�20

�in

33�

x3

�20

�in

317

5��

4

3x

4x

5x

7x

� � 122 � 3

3x — 2

x

a

x

x

b

x

2x5x

19 � 2� � 2

x � 2

3x2x

2xx

yx

xy

x

x

r

h

r

h

r

1 � 3

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-5

8-5



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Mu

ltip

lyin

g a

Po

lyn

om

ial b

y a

Mo

no

mia

l

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill48

5G

lenc

oe A

lgeb

ra 1

Lesson 8-6

Pro

du

ct o

f M

on

om

ial a

nd

Po

lyn

om

ial

Th

e D

istr

ibu

tive

Pro

pert

y ca

n b

e u

sed

tom

ult

iply

a p

olyn

omia

l by

a m

onom

ial.

You

can

mu

ltip

ly h

oriz

onta

lly

or v

erti

call

y.S

omet

imes

mu

ltip

lyin

g re

sult

s in

lik

e te

rms.

Th

e pr

odu

cts

can

be

sim

plif

ied

by c

ombi

nin

g li

ke t

erm

s.

Fin

d �

3x2 (

4x2

�6x

�8)

.

Hor

izon

tal

Met

hod

�3x

2 (4x

2�

6x�

8)�

�3x

2 (4x

2 ) �

(�3x

2 )(6

x) �

(�3x

2 )(8

)�

�12

x4�

(�18

x3)

�(�

24x2

)�

�12

x4�

18x3

�24

x2

Ver

tica

l M

eth

od4x

2�

6x�

8(�

) �

3x2

�12

x4�

18x3

�24

x2

Th

e pr

odu

ct i

s �

12x4

�18

x3�

24x2

.

Sim

pli

fy �

2(4x

2�

5x)

�x

(x2

�6x

).

�2(

4x2

�5x

) �

x( x

2�

6x)

��

2(4x

2 ) �

(�2)

(5x)

�(�

x)(x

2 ) �

(�x)

(6x)

��

8x2

�(�

10x)

�(�

x3)

�(�

6x2 )

�(�

x3)

�[�

8x2

�(�

6x2 )

] �

(�10

x)�

�x3

�14

x2�

10x

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d e

ach

pro

du

ct.

1.x(

5x�

x2)

2.x(

4x2

�3x

�2)

3.�

2xy(

2y�

4x2 )

5x2 �

x3

4x3 �

3x2 �

2x�

4xy

2�

8x3 y

4.�

2g(g

2�

2g�

2)5.

3x(x

4�

x3�

x2)

6.�

4x(2

x3�

2x�

3)

�2g

3�

4g2

�4g

3x5

�3x

4�

3x3

�8x

4 �8x

2 �12

x

7.�

4cx(

10 �

3x)

8.3y

(�4x

�6x

3 �2y

)9.

2x2 y

2 (3x

y�

2y�

5x)

�40

cx�

12cx

2�

12xy

�18

x3 y

�6y

26x

3 y3

�4x

2 y3

�10

x3 y

2

Sim

pli

fy.

10.x

(3x

�4)

�5x

11.�

x(2x

2�

4x)

�6x

2

3x2

�9x

�2x

3�

2x2

12.6

a(2a

�b)

�2a

(�4a

�5b

)13

.4r(

2r2

�3r

�5)

�6r

(4r2

�2r

�8)

4a2

�4a

b32

r3�

68r

14.4

n(3

n2

�n

�4)

�n

(3 �

n)

15.2

b(b2

�4b

�8)

�3b

(3b2

�9b

�18

)

12n

3�

5n2

�19

n�

7b3

�19

b2

�70

b

16.�

2z(4

z2�

3z�

1) �

z(3z

2�

2z�

1)17

.2(4

x2�

2x)

�3(

�6x

2�

4) �

2x(x

�1)

�11

z3

�4z

2�

z28

x2

�6x

�12

©G

lenc

oe/M

cGra

w-H

ill48

6G

lenc

oe A

lgeb

ra 1

Solv

e Eq

uat

ion

s w

ith

Po

lyn

om

ial E

xpre

ssio

ns

Man

y eq

uat

ion

s co

nta

inpo

lyn

omia

ls t

hat

mu

st b

e ad

ded,

subt

ract

ed,o

r m

ult

ipli

ed b

efor

e th

e eq

uat

ion

can

be

solv

ed.

Sol

ve 4

(n�

2) �

5n�

6(3

�n

) �

19.

4(n

�2)

�5n

�6(

3 �

n)

�19

Orig

inal

equ

atio

n

4n�

8 �

5n�

18 �

6n�

19D

istr

ibut

ive

Pro

pert

y

9n�

8 �

37 �

6nC

ombi

ne li

ke t

erm

s.

15n

�8

�37

Add

6n

to b

oth

side

s.

15n

�45

Add

8 t

o bo

th s

ides

.

n�

3D

ivid

e ea

ch s

ide

by 1

5.

Th

e so

luti

on i

s 3.

Sol

ve e

ach

eq

uat

ion

.

1.2(

a�

3) �

3(�

2a�

6)3

2.3(

x�

5) �

6 �

183

3.3x

(x�

5) �

3x2

��

302

4.6(

x2�

2x)

�2(

3x2

�12

)2

5.4(

2p�

1) �

12p

�2(

8p�

12)

�1

6.2(

6x�

4) �

2 �

4(x

�4)

�3

7.�

2(4y

�3)

�8y

�6

�4(

y�

2)1

8.c(

c�

2) �

c(c

�6)

�10

c�

126

9.3(

x2�

2x)

�3x

2�

5x�

111

10.2

(4x

�3)

�2

��

4(x

�1)

�1

11.3

(2h

�6)

�(2

h�

1) �

97

12.3

(y�

5) �

(4y

�8)

��

2y�

10�

13

13.3

(2a

�6)

�(�

3a�

1) �

4a�

22

14.5

(2x2

�1)

�(1

0x2

�6)

��

(x�

2)�

3

15.3

(x�

2) �

2(x

�1)

��

5(x

�3)

16.4

(3p2

�2p

) �

12p2

�2(

8p�

6)�

3 � 27 � 10

1 �

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Mu

ltip

lyin

g a

Po

lyn

om

ial b

y a

Mo

no

mia

l

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises

3 � 5



Skil

ls P

ract

ice

Mu

ltip

lyin

g a

Po

lyn

om

ial b

y a

Mo

no

mia

l

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill48

7G

lenc

oe A

lgeb

ra 1

Lesson 8-6

Fin

d e

ach

pro

du

ct.

1.a(

4a�

3)2.

�c(

11c

�4)

4a2

3a

�11

c2

�4c

3.x(

2x�

5)4.

2y(y

�4)

2x2

�5x

2y2

�8y

5.�

3n(n

2�

2n)

6.4h

(3h

�5)

�3n

3�

6n2

12h

2�

20h

7.3x

(5x2

�x

�4)

8.7c

(5 �

2c2

�c3

)

15x3

�3x

2

12x

35c

�14

c3

7c4

9.�

4b(1

�9b

�2b

2 )10

.6y(

�5

�y

�4y

2 )

�4b

36

b2

8b

3�

30y

�6y

2

24y

3

11.2

m2 (

2m2

�3m

�5)

12.�

3n2 (

�2n

2�

3n�

4)

4m4

6m

3�

10m

26n

4�

9n3

�12

n2

Sim

pli

fy.

13.w

(3w

�2)

�5w

14.f

(5f

�3)

�2f

3w2

7w

5f2

�5f

15.�

p(2p

�8)

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b(�

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eq

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5 �

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4(x

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23.5

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Fin

d e

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pli

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23.N

UM

BER

TH

EORY

Let

xbe

an

in

tege

r.W

hat

is

the

prod

uct

of

twic

e th

e in

tege

r ad

ded

to t

hre

e ti

mes

th

e n

ext

con

secu

tive

in

tege

r?5x

3

INV

ESTM

ENTS

For

Exe

rcis

es 2

4–26

,use

th

e fo

llow

ing

info

rmat

ion

.K

ent

inve

sted

$5,

000

in a

ret

irem

ent

plan

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allo

cate

d x

doll

ars

of t

he

mon

ey t

o a

bon

dac

cou

nt

that

ear

ns

4% i

nte

rest

per

yea

r an

d th

e re

st t

o a

trad

itio

nal

acc

oun

t th

at e

arn

s 5%

inte

rest

per

yea

r.

24.W

rite

an

exp

ress

ion

th

at r

epre

sen

ts t

he

amou

nt

of m

oney

in

vest

ed i

n t

he

trad

itio

nal

acco

un

t.5,

000

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25.W

rite

a p

olyn

omia

l m

odel

in

sim

ples

t fo

rm f

or t

he

tota

l am

oun

t of

mon

ey T

Ken

t h

asin

vest

ed a

fter

on

e ye

ar.(

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t:E

ach

acc

oun

t h

as A

�IA

doll

ars,

wh

ere

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th

e or

igin

alam

oun

t in

th

e ac

cou

nt

and

Iis

its

in

tere

st r

ate.

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�5,

250

�0.

01x

26.I

f K

ent

put

$500

in

th

e bo

nd

acco

un

t,h

ow m

uch

mon

ey d

oes

he

hav

e in

his

ret

irem

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plan

aft

er o

ne

year

?$5

,245

3 � 2

1 � 41 � 2

7 � 41 � 4

2 � 31 � 4

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rag

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Mu

ltip

lyin

g a

Po

lyn

om

ial b

y a

Mo

no

mia

l

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csM

ult

iply

ing

a P

oly

no

mia

l by

a M

on

om

ial

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6

©G

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w-H

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9G

lenc

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lgeb

ra 1

Lesson 8-6

Pre-

Act

ivit

yH

ow i

s fi

nd

ing

the

pro

du

ct o

f a

mon

omia

l an

d a

pol

ynom

ial

rela

ted

to f

ind

ing

the

area

of

a re

ctan

gle?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-6

at

the

top

of p

age

444

in y

our

text

book

.

You

may

rec

all

that

th

e fo

rmu

la f

or t

he

area

of

a re

ctan

gle

is A

��w

.In

this

rec

tan

gle,

��

and

w�

.How

wou

ld y

ou

subs

titu

te t

hes

e va

lues

in

th

e ar

ea f

orm

ula

?

A�

(x

3)(2

x)

Rea

din

g t

he

Less

on

1.R

efer

to

Les

son

8-6

.

a.H

ow i

s th

e D

istr

ibu

tive

Pro

pert

y u

sed

to m

ult

iply

a p

olyn

omia

l by

a m

onom

ial?

Th

e m

on

om

ial i

s m

ult

iplie

d b

y ea

ch t

erm

in t

he

po

lyn

om

ial.

b.

Use

th

e D

istr

ibu

tive

Pro

pert

y to

com

plet

e th

e fo

llow

ing.

2y2 (

3y2

�2y

�7)

�2y

2 ()

�2y

2 ()

�2y

2 ()

��

�

�3x

3 (x3

�2x

2�

3)�

��

��

�

2.W

hat

is

the

diff

eren

ce b

etw

een

sim

plif

yin

g an

exp

ress

ion

an

d so

lvin

g an

equ

atio

n?

Sim

plif

yin

g a

n e

xpre

ssio

n is

co

mb

inin

g li

ke t

erm

s.S

olv

ing

an

eq

uat

ion

is f

ind

ing

th

e va

lue

of

the

vari

able

th

at m

akes

th

e eq

uat

ion

tru

e.

Hel

pin

g Y

ou

Rem

emb

er

3.U

se t

he

equ

atio

n 2

x(x

�5)

�3x

(x�

3) �

5x(x

�7)

�9

to s

how

how

you

wou

ld e

xpla

inth

e pr

oces

s of

sol

vin

g eq

uat

ion

s w

ith

pol

ynom

ial

expr

essi

ons

to a

not

her

alg

ebra

stu

den

t.

Use

th

e D

istr

ibu

tive

Pro

per

ty.

2x2

�10

x

3x2

9x

�5x

2

35x

�9

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mb

ine

like

term

s.5 x

2�

x�

5x2

35

x�

9

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btr

act

5 x2

fro

m b

oth

sid

es.

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x�

9

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btr

act

35x

fro

m b

oth

sid

es.

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x�

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ide

each

sid

e by

�36

.x

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25

9x3

6x5

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6

(�3x

3 )(3

)(�

3x3 )

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2 )�

3x3 (

x3 )

14y

24y

36y

4

72y

3y2

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3

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Fig

ura

te N

um

ber

sT

he

nu

mbe

rs b

elow

are

cal

led

pen

tago

nal

nu

mb

ers.

Th

ey a

re t

he

nu

mbe

rs o

f do

ts o

r di

sks

that

can

be

arra

nge

d as

pen

tago

ns.

1.F

ind

the

prod

uct

n

(3n

�1)

.�

2.E

valu

ate

the

prod

uct

in

Exe

rcis

e 1

for

valu

es o

f n

from

1 t

hro

ugh

4.

1,5,

12,2

2

3.W

hat

do

you

not

ice?

Th

ey a

re t

he

firs

t fo

ur

pen

tag

on

al n

um

ber

s.

4.F

ind

the

nex

t si

x pe

nta

gon

al n

um

bers

.35

,51,

70,9

2,11

7,14

5

5.F

ind

the

prod

uct

n

(n�

1).

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valu

ate

the

prod

uct

in

Exe

rcis

e 5

for

valu

es o

f n

from

1 t

hro

ugh

5.O

n a

not

her

sh

eet

ofpa

per,

mak

e dr

awin

gs t

o sh

ow w

hy

thes

e n

um

bers

are

cal

led

the

tria

ngu

lar

nu

mbe

rs.

1,3,

6,10

,15

7.F

ind

the

prod

uct

n(2

n�

1).

2n2

�n

8.E

valu

ate

the

prod

uct

in

Exe

rcis

e 7

for

valu

es o

f n

from

1 t

hro

ugh

5.D

raw

th

ese

hex

agon

al n

um

bers

.1,

6,15

,28,

45

9.F

ind

the

firs

t 5

squ

are

nu

mbe

rs.A

lso,

wri

te t

he

gen

eral

exp

ress

ion

for

an

y sq

uar

en

um

ber.

1,4,

9,16

,25;

n2

Th

e n

um

bers

you

hav

e ex

plor

ed a

bove

are

cal

led

the

plan

e fi

gura

te n

um

bers

bec

ause

th

eyca

n b

e ar

ran

ged

to m

ake

geom

etri

c fi

gure

s.Yo

u c

an a

lso

crea

te s

olid

fig

ura

te n

um

bers

.

10.I

f yo

u p

ile

10 o

ran

ges

into

a p

yram

id w

ith

a t

rian

gle

as a

bas

e,yo

u g

et o

ne

of t

he

tetr

ahed

ral

nu

mbe

rs.H

ow m

any

laye

rs a

re t

her

e in

th

e py

ram

id?

How

man

y or

ange

sar

e th

ere

in t

he

bott

om l

ayer

s?3

laye

rs;

6

11.E

valu

ate

the

expr

essi

on

n3

�n

2�

nfo

r va

lues

of

nfr

om 1

th

rou

gh 5

to

fin

d th

e

firs

t fi

ve t

etra

hed

ral

nu

mbe

rs.

1,4,

10,2

0,35

1 � 31 � 2

1 � 6

n � 2n

2� 2

1 � 2

n � 23 n

2�

21 � 2

●●

2212

51

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-6

8-6



Stu

dy G

uid

e a

nd I

nte

rven

tion

Mu

ltip

lyin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

©G

lenc

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w-H

ill49

1G

lenc

oe A

lgeb

ra 1

Lesson 8-7

Mu

ltip

ly B

ino

mia

lsT

o m

ult

iply

tw

o bi

nom

ials

,you

can

app

ly t

he

Dis

trib

uti

ve P

rope

rty

twic

e.A

use

ful

way

to

keep

tra

ck o

f te

rms

in t

he

prod

uct

is

to u

se t

he

FO

IL m

eth

od a

sil

lust

rate

d in

Exa

mpl

e 2.

Fin

d (

x

3)(x

�4)

.

Hor

izon

tal

Met

hod

(x�

3)(x

�4)

�x(

x�

4) �

3(x

�4)

�(x

)(x)

�x(

�4)

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x)�

3(�

4)�

x2�

4x�

3x�

12�

x2�

x�

12

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tica

l M

eth

od

x�

3(�

) x

�4

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�12

x2�

3xx2

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�12

Th

e pr

odu

ct i

s x2

�x

�12

.

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d (

x�

2)(x

5)

usi

ng

the

FO

IL m

eth

od.

(x�

2)(x

�5)

Firs

t O

uter

In

ner

Last

�(x

)(x)

�(x

)(5)

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2)(x

) �

(�2)

(5)

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�(�

2x)

�10

�x2

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�10

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e pr

odu

ct i

s x2

�3x

�10

.

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ple1

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ple1

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ple2

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ple2

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cises

Exer

cises

Fin

d e

ach

pro

du

ct.

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�2)

(x�

3)2.

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12

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lenc

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Mu

ltip

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oly

no

mia

lsT

he

Dis

trib

uti

ve P

rope

rty

can

be

use

d to

mu

ltip

ly a

ny

two

poly

nom

ials

. Fin

d (

3x

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x2�

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e pr

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d e

ach

pro

du

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3n

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2t4

7t

3�

9t2

�11

t

3

13.(

y2�

5y�

3)(2

y2�

7y�

4)14

.(3b

2�

2b�

1)(2

b2�

3b�

4)

2y4

�3y

3�

33y

2

41y

�12

6b4

�13

b3

�4b

2

5b�

4

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Mu

ltip

lyin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Mu

ltip

lyin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill49

3G

lenc

oe A

lgeb

ra 1

Lesson 8-7

Fin

d e

ach

pro

du

ct.

1.(m

�4)

(m�

1)2.

(x�

2)(x

�2)

m2

5m

4

x2

4x

4

3.(b

�3)

(b�

4)4.

(t�

4)(t

�3)

b2

7b

12

t2

t�

12

5.(r

�1)

(r�

2)6.

(z�

5)(z

�1)

r2�

r�

2z2

�4z

�5

7.(3

c�

1)(c

�2)

8.(2

x�

6)(x

�3)

3c2

�5c

�2

2x2

�18

9.(d

�1)

(5d

�4)

10.(

2��

5)(�

�4)

5d2

�9d

4

2�2

�3�

�20

11.(

3n�

7)(n

�3)

12.(

q�

5)(5

q�

1)

3n2

2n

�21

5q2

24

q�

5

13.(

3b�

3)(3

b�

2)14

.(2m

�2)

(3m

�3)

9b2

3b

�6

6m2

�6

15.(

4c�

1)(2

c�

1)16

.(5a

�2)

(2a

�3)

8c2

6c

1

10a2

�19

a

6

17.(

4h�

2)(4

h�

1)18

.(x

�y)

(2x

�y)

16h

2�

12h

2

2x2

�3x

y

y2

19.(

e�

4)(e

2�

3e�

6)20

.(t

�1)

(t2

�2t

�4)

e3

7e2

6e

�24

t3

3t2

6t

4

21.(

k�

4)(k

2�

3k�

6)22

.(m

�3)

(m2

�3m

�5)

k3

7k

2

6k�

24m

3

6m2

14

m

15

GE

OM

ET

RY

Wri

te a

n e

xpre

ssio

n t

o re

pre

sen

t th

e ar

ea o

f ea

ch f

igu

re.

23.

24.

8x2

18

x�

5 u

nit

s2(x

2

4x

4)�

un

its2

2x

4

2x

5

4x �

1

©G

lenc

oe/M

cGra

w-H

ill49

4G

lenc

oe A

lgeb

ra 1

Fin

d e

ach

pro

du

ct.

1.(q

�6)

(q�

5)2.

(x�

7)(x

�4)

3.(s

�5)

(s�

6)q

2

11q

30

x2

11x

28

s2�

s�

30

4.(n

�4)

(n�

6)5.

(a�

5)(a

�8)

6.(w

�6)

(w�

9)n

2�

10n

24

a2�

13a

40

w2

�15

w

54

7.(4

c�

6)(c

�4)

8.(2

x�

9)(2

x�

4)9.

(4d

�5)

(2d

�3)

4c2

�10

c�

244x

2�

10x

�36

8d2

�22

d

15

10.(

4b�

3)(3

b�

4)11

.(4m

�2)

(4m

�3)

12.(

5c�

5)(7

c�

9)12

b2

�7b

�12

16m

2�

4m�

635

c2

10

c�

45

13.(

6a�

3)(7

a�

4)14

.(6h

�3)

(4h

�2)

15.(

2x�

2)(5

x�

4)42

a2

�45

a

1224

h2

�24

h

610

x2�

18x

8

16.(

3a�

b)(2

a�

b)17

.(4g

�3h

)(2g

�3h

)18

.(4x

�y)

(4x

�y)

6a2

�5a

b

b2

8g2

18

gh

9h

216

x2

8xy

y

2

19.(

m�

5)(m

2�

4m�

8)20

.(t

�3)

(t2

�4t

�7)

m3

9m

2

12m

�40

t3

7t2

19

t

21

21.(

2h�

3)(2

h2

�3h

�4)

22.(

3d�

3)(2

d2

�5d

�2)

4h3

12

h2

17

h

126d

3

21d

2

9d�

6

23.(

3q�

2)(9

q2�

12q

�4)

24.(

3r�

2)(9

r2�

6r�

4)27

q3

�18

q2

�12

q

827

r3

36r2

24

r

8

25.(

3c2

�2c

�1)

(2c2

�c

�9)

26.(

2�2

��

�3)

(4�2

�2�

�2)

6c4

7c

3

27c2

17

c�

98�

4

8�3

10

�2

4��

6

27.(

2x2

�2x

�3)

(2x2

�4x

�3)

28.(

3y2

�2y

�2)

(3y2

�4y

�5)

4x4

�12

x3

8x

2

6x�

99y

4�

6y3

�17

y2

�18

y�

10

GEO

MET

RYW

rite

an

exp

ress

ion

to

rep

rese

nt

the

area

of

each

fig

ure

.

29.

4x2

�2x

�2

un

its2

30.

4x2

3x

�1

un

its2

31.N

UM

BER

TH

EORY

Let

xbe

an

eve

n i

nte

ger.

Wh

at i

s th

e pr

odu

ct o

f th

e n

ext

two

con

secu

tive

eve

n i

nte

gers

?x2

6x

8

32.G

EOM

ETRY

Th

e vo

lum

e of

a r

ecta

ngu

lar

pyra

mid

is

one

thir

d th

e pr

odu

ct o

f th

e ar

eaof

its

bas

e an

d it

s h

eigh

t.F

ind

an e

xpre

ssio

n f

or t

he

volu

me

of a

rec

tan

gula

r py

ram

idw

hos

e ba

se h

as a

n a

rea

of 3

x2�

12x

�9

squ

are

feet

an

d w

hos

e h

eigh

t is

x�

3 fe

et.

x3

7x2

15

x

9 fe

et3

3x

2

5x �

4

x

1

4x

2

2x �

2

Pra

ctic

e (

Ave

rag

e)

Mu

ltip

lyin

g P

oly

no

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7



Readin

g t

o L

earn

Math

em

ati

csM

ult

iply

ing

Po

lyn

om

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill49

5G

lenc

oe A

lgeb

ra 1

Lesson 8-7

Pre-

Act

ivit

yH

ow i

s m

ult

iply

ing

bin

omia

ls s

imil

ar t

o m

ult

iply

ing

two-

dig

itn

um

ber

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-7

at

the

top

of p

age

452

in y

our

text

book

.

In y

our

own

wor

ds,e

xpla

in h

ow t

he

dist

ribu

tive

pro

pert

y is

use

d tw

ice

tom

ult

iply

tw

o-di

git

nu

mbe

rs.

Th

e o

nes

of

the

firs

t fa

cto

r ar

e m

ult

iplie

d b

y th

e te

ns

and

th

eo

nes

of

the

oth

er f

acto

r.T

hen

th

e te

ns

of

the

firs

t fa

cto

r ar

em

ult

iplie

d b

y th

e te

ns

and

on

es o

f th

e o

ther

fac

tor.

Rea

din

g t

he

Less

on

1.H

ow i

s m

ult

iply

ing

bin

omia

ls s

imil

ar t

o m

ult

iply

ing

two-

digi

t n

um

bers

?

Bin

om

ials

hav

e tw

o t

erm

s an

d e

ach

ter

m o

f o

ne

bin

om

ial i

s m

ult

iplie

d b

yea

ch t

erm

of

the

oth

er b

ino

mia

l.

2.C

ompl

ete

the

tabl

e u

sin

g th

e F

OIL

met

hod

.

Pro

du

ct o

f

Pro

du

ct o

f

Pro

du

ct o

f

Pro

du

ct o

f F

irst

Ter

ms

Ou

ter T

erm

sIn

ner

Ter

ms

Las

t Ter

ms

(x�

5)(x

�3)

�(x

)(x)

�(x

)(�

3)�

(5)(

x)�

(5)(

�3)

�x2

��

3x�

5x�

�15

�x2

�2x

�15

(3y

�6)

(y�

2)�

(3y)

(y)

�(3

y)(�

2)�

(6)(

y)�

(6)(

�2)

�3y

2�

�6y

�6y

��

12

�3y

2�

12

Hel

pin

g Y

ou

Rem

emb

er

3.T

hin

k of

a m

eth

od f

or r

emem

beri

ng

all

the

prod

uct

com

bin

atio

ns

use

d in

th

e F

OIL

met

hod

for

mu

ltip

lyin

g tw

o bi

nom

ials

.Des

crib

e yo

ur

met

hod

usi

ng

wor

ds o

r a

diag

ram

.S

amp

le a

nsw

er:

Imag

ine

that

th

e tw

o b

ino

mia

ls a

re w

ritt

en o

n t

he

flo

or.

Fo

r F

OIL

,th

ink

of

all t

he

po

ssib

le w

ays

you

co

uld

hav

e yo

ur

left

fo

ot

on

a te

rm o

f th

e fi

rst

bin

om

ial a

nd

yo

ur

rig

ht

foo

t o

n a

ter

m o

f th

e se

con

db

ino

mia

l.Yo

ur

feet

co

uld

be

on

th

e fi

rst

term

s,th

e o

ute

r te

rms,

the

inn

erte

rms,

or

the

last

ter

ms.

©G

lenc

oe/M

cGra

w-H

ill49

6G

lenc

oe A

lgeb

ra 1

Pas

cal’s

Tri

ang

leT

his

arr

ange

men

t of

nu

mb

ers

is c

alle

d P

asca

l’s T

rian

gle.

It w

as f

irst

pu

bli

shed

in

166

5,b

ut

was

kn

own

hu

nd

red

s of

yea

rs e

arli

er.

1.E

ach

nu

mbe

r in

th

e tr

ian

gle

is f

oun

d by

add

ing

two

nu

mbe

rs.

Wh

at t

wo

nu

mbe

rs w

ere

adde

d to

get

th

e 6

in t

he

5th

row

?

3 an

d 3

2.D

escr

ibe

how

to

crea

te t

he

6th

row

of

Pas

cal’s

Tri

angl

e.

Th

e fi

rst

and

last

nu

mb

ers

are

1.E

valu

ate

1

4,4

6,

6

4,an

d 4

1

to f

ind

th

e o

ther

nu

mb

ers.

3.W

rite

th

e n

um

bers

for

row

s 6

thro

ugh

10

of t

he

tria

ngl

e.

Row

6:

15

1010

51

Row

7:

16

1520

156

1R

ow 8

:1

721

3535

217

1R

ow 9

:1

828

5670

5628

81

Row

10:

19

3684

126

126

8436

91

Mu

ltip

ly t

o fi

nd

th

e ex

pan

ded

for

m o

f ea

ch p

rod

uct

.

4.(a

�b)

2a2

2a

b

b2

5.(a

�b)

3a3

3a

2 b

3ab

2

b3

6.(a

�b)

4a4

4a

3 b

6a2 b

2

4ab

3

b4

Now

com

par

e th

e co

effi

cien

ts o

f th

e th

ree

pro

du

cts

in E

xerc

ises

4–6

wit

h

Pas

cal’s

Tri

angl

e.

7.D

escr

ibe

the

rela

tion

ship

bet

wee

n t

he

expa

nde

d fo

rm o

f (a

�b)

nan

d P

asca

l’s T

rian

gle.

Th

e co

effi

cien

ts o

f th

e ex

pan

ded

fo

rm a

re f

ou

nd

in r

ow

n

1 o

f P

asca

l’sTr

ian

gle

.

8.U

se P

asca

l’s T

rian

gle

to w

rite

th

e ex

pan

ded

form

of

(a�

b)6 .

a6

6a5 b

15

a4b

2

20a3

b3

15

a2b

4

6ab

5

b6

11

11

21

13

31

14

64

1

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-7

8-7



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Sp

ecia

l Pro

du

cts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-8

8-8

©G

lenc

oe/M

cGra

w-H

ill49

7G

lenc

oe A

lgeb

ra 1

Lesson 8-8

Squ

ares

of

Sum

s an

d D

iffe

ren

ces

Som

e pa

irs

of b

inom

ials

hav

e pr

odu

cts

that

foll

ow s

peci

fic

patt

ern

s.O

ne

such

pat

tern

is

call

ed t

he

squ

are

of a

su

m.A

not

her

is

call

ed t

he

squ

are

of a

dif

fere

nce

.

Sq

uar

e o

f a

sum

(a�

b)2

�(a

�b)

(a�

b) �

a2

�2a

b�

b2

Sq

uar

e o

f a

dif

fere

nce

(a�

b)2

�(a

�b)

(a�

b) �

a2

�2a

b�

b2

Fin

d (

3a

4)(3

a

4).

Use

th

e sq

uar

e of

a s

um

pat

tern

,wit

h a

�3a

and

b�

4.

(3a

�4)

(3a

�4)

�(3

a)2

�2(

3a)(

4) �

(4)2

�9a

2�

24a

�16

Th

e pr

odu

ct i

s 9a

2�

24a

�16

.

Fin

d (

2z�

9)(2

z�

9).

Use

th

e sq

uar

e of

a d

iffe

ren

ce p

atte

rn w

ith

a�

2zan

d b

�9.

(2z

�9)

(2z

�9)

�(2

z)2

�2(

2z)(

9) �

(9)(

9)�

4z2

�36

z�

81

Th

e pr

odu

ct i

s 4z

2�

36z

�81

.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d e

ach

pro

du

ct.

1.(x

�6)

22.

(3p

�4)

23.

(4x

�5)

2

x2

�12

x

369p

2

24p

16

16x2

�40

x

25

4.(2

x�

1)2

5.(2

h�

3)2

6.(m

�5)

2

4x2

�4x

1

4h2

12

h

9m

2

10m

25

7.(c

�3)

28.

(3 �

p)2

9.(x

�5y

)2

c2

6c

9

9 �

6p

p2

x2�

10xy

25

y2

10.(

8y�

4)2

11.(

8 �

x)2

12.(

3a�

2b)2

64y

2

64y

16

64

16x

x2

9a2

�12

ab

4b2

13.(

2x�

8)2

14.(

x2�

1)2

15.(

m2

�2)

2

4x2

�32

x

64x

4

2x2

1

m4

�4m

2

4

16.(

x3�

1)2

17.(

2h2

�k2

)218

. �x

�3 �2

x6

�2x

3 1

4h4

�4h

2 k2

k

4x2

x

9

19.(

x�

4y2 )

220

.(2p

�4q

)221

. �x

�2 �2

x2�

8xy

2

16y

44p

2

16p

q

16q

2x2

�x

4

8 � 34 � 92 � 3

3 � 21 � 161 � 4

©G

lenc

oe/M

cGra

w-H

ill49

8G

lenc

oe A

lgeb

ra 1

Pro

du

ct o

f a

Sum

an

d a

Dif

fere

nce

Th

ere

is a

lso

a pa

tter

n f

or t

he

prod

uct

of

asu

m a

nd

a di

ffer

ence

of

the

sam

e tw

o te

rms,

(a�

b)(a

�b)

.Th

e pr

odu

ct i

s ca

lled

th

ed

iffe

ren

ce o

f sq

uar

es.

Pro

du

ct o

f a

Su

m a

nd

a D

iffe

ren

ce(a

�b)

(a�

b) �

a2�

b2

Fin

d (

5x

3y)(

5x�

3y).

(a�

b)(a

�b)

�a2

�b2

Pro

duct

of

a S

um a

nd a

Diff

eren

ce

(5x

�3y

)(5x

�3y

) �

(5x)

2�

(3y)

2a

�5x

and

b�

3y

�25

x2�

9y2

Sim

plify

.

Th

e pr

odu

ct i

s 25

x2�

9y2 .

Fin

d e

ach

pro

du

ct.

1.(x

�4)

(x�

4)2.

(p�

2)(p

�2)

3.(4

x�

5)(4

x�

5)

x2�

16p

2�

416

x2�

25

4.(2

x�

1)(2

x�

1)5.

(h�

7)(h

�7)

6.(m

�5)

(m�

5)

4x2

�1

h2

�49

m2

�25

7.(2

c�

3)(2

c�

3)8.

(3 �

5q)(

3 �

5q)

9.(x

�y)

(x�

y)

4c2

�9

9 �

25q

2x2

�y2

10.(

y�

4x)(

y�

4x)

11.(

8 �

4x)(

8 �

4x)

12.(

3a�

2b)(

3a�

2b)

y2�

16x2

64 �

16x2

9a2

�4b

2

13.(

3y�

8)(3

y�

8)14

.(x2

�1)

(x2

�1)

15.(

m2

�5)

(m2

�5)

9y2

�64

x4

�1

m4

�25

16.(

x3�

2)(x

3�

2)17

.(h

2�

k2)(

h2

�k2

)18

. �x

�2 ��

x�

2 �x

6�

4h

4�

k4

x2�

4

19.(

3x�

2y2 )

(3x

�2y

2 )20

.(2p

�5s

)(2p

�5s

)21

. �x

�2y

��x

�2y

�9x

2�

4y4

4p2

�25

s2x2

�4y

216 � 9

4 � 34 � 31 � 16

1 � 41 � 4

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Sp

ecia

l Pro

du

cts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-8

8-8

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Sp

ecia

l Pro

du

cts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

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ER

IOD

____

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8-8

8-8

©G

lenc

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cGra

w-H

ill49

9G

lenc

oe A

lgeb

ra 1

Lesson 8-8

Fin

d e

ach

pro

du

ct.

1.(n

�3)

22.

(x�

4)(x

�4)

n2

6n

9

x2

8x

16

3.(y

�7)

24.

(t�

3)(t

�3)

y2

�14

y

49t2

�6t

9

5.(b

�1)

(b�

1)6.

(a�

5)(a

�5)

b2

�1

a2�

25

7.(p

�4)

28.

(z�

3)(z

�3)

p2

�8p

16

z2�

9

9.(�

�2)

(��

2)10

.(r

�1)

(r�

1)

�2

4�

4r2

�2r

1

11.(

3g�

2)(3

g�

2)12

.(2m

�3)

(2m

�3)

9g2

�4

4m2

�9

13.(

6 �

u)2

14.(

r�

s)2

36

12u

u

2r2

2r

s

s2

15.(

3q�

1)(3

q�

1)16

.(c

�e)

2

9q2

�1

c2�

2ce

e

2

17.(

2k�

2)2

18.(

w�

3h)2

4k2

�8k

4

w2

6w

h

9h2

19.(

3p�

4)(3

p�

4)20

.(t

�2u

)2

9p2

� 1

6t2

4t

u

4u2

21.(

x�

4y)2

22.(

3b�

7)(3

b�

7)

x2�

8xy

16

y2

9b2

�49

23.(

3y�

3g)(

3y�

3g)

24.(

s2�

r2)2

9y2

�9g

2s4

2s

2 r2

r4

25.(

2k�

m2 )

226

.(3u

2�

n)2

4k2

4k

m2

m

49u

4�

6u2 n

n

2

27.G

EOM

ETRY

Th

e le

ngt

h o

f a

rect

angl

e is

th

e su

m o

f tw

o w

hol

e n

um

bers

.Th

e w

idth

of

the

rect

angl

e is

th

e di

ffer

ence

of

the

sam

e tw

o w

hol

e n

um

bers

.Usi

ng

thes

e fa

cts,

wri

te a

verb

al e

xpre

ssio

n f

or t

he

area

of

the

rect

angl

e.T

he

area

is t

he

squ

are

of

the

larg

er n

um

ber

min

us

the

squ

are

of

the

smal

ler

nu

mb

er.

©G

lenc

oe/M

cGra

w-H

ill50

0G

lenc

oe A

lgeb

ra 1

Fin

d e

ach

pro

du

ct.

1.(n

�9)

22.

(q�

8)2

3.(�

�10

)2

n2

18

n

81q

2

16q

64

�2�

20�

10

0

4.(r

�11

)25.

(p�

7)2

6.(b

�6)

(b�

6)r2

�22

r

121

p2

14

p

49b

2�

36

7.(z

�13

)(z

�13

)8.

(4e

�2)

29.

(5w

�4)

2

z2�

169

16e

2

16e

4

25w

2�

40w

16

10.(

6h�

1)2

11.(

3s�

4)2

12.(

7v�

2)2

36h

2�

12h

1

9s2

24

s

1649

v2

�28

v

4

13.(

7k�

3)(7

k�

3)14

.(4d

�7)

(4d

�7)

15.(

3g�

9h)(

3g�

9h)

49k

2�

916

d2

�49

9g2

�81

h2

16.(

4q�

5t)(

4q�

5t)

17.(

a�

6u)2

18.(

5r�

s)2

16q

2�

25t2

a2

12

au

36u

225

r2

10rs

s2

19.(

6c�

m)2

20.(

k�

6y)2

21.(

u�

7p)2

36c

2�

12cm

m

2k2

�12

ky

36y

2u

2�

14u

p

49p

2

22.(

4b�

7v)2

23.(

6n�

4p)2

24.(

5q�

6s)2

16b

2�

56bv

49

v2

36n

2

48n

p

16p

225

q2

60

qs

36

s2

25.(

6a�

7b)(

6a�

7b)

26.(

8h�

3d)(

8h�

3d)

27.(

9x�

2y2 )

2

36a2

�49

b2

64h

2�

9d2

81x2

36

xy2

4y

4

28.(

3p3

�2m

)229

.(5a

2�

2b)2

30.(

4m3

�2t

)2

9p6

12

p3 m

4m

225

a4�

20a2

b

4b2

16m

6�

16m

3 t

4t2

31.(

6e3

�c)

232

.(2b

2�

g)(2

b2�

g)33

.(2v

2�

3e2 )

(2v2

�3e

2 )36

e6�

12e

3 c

c2

4b4

�g

24v

4

12v2

e2

9e4

34.G

EOM

ETRY

Jan

elle

wan

ts t

o en

larg

e a

squ

are

grap

h t

hat

sh

e h

as m

ade

so t

hat

a s

ide

of t

he

new

gra

ph w

ill

be 1

in

ch m

ore

than

tw

ice

the

orig

inal

sid

e s.

Wh

at t

rin

omia

lre

pres

ents

th

e ar

ea o

f th

e en

larg

ed g

raph

?4s

2

4s

1

GEN

ETIC

SF

or E

xerc

ises

35

and

36,

use

th

e fo

llow

ing

info

rmat

ion

.In

a g

uin

ea p

ig,p

ure

bla

ck h

air

colo

rin

g B

is d

omin

ant

over

pu

re w

hit

e co

lori

ng

b.S

upp

ose

two

hyb

rid

Bb

guin

ea p

igs,

wit

h b

lack

hai

r co

lori

ng,

are

bred

.

35.F

ind

an e

xpre

ssio

n f

or t

he

gen

etic

mak

e-u

p of

th

e gu

inea

pig

off

spri

ng.

0.25

BB

0.

50B

b

0.25

bb

36.W

hat

is

the

prob

abil

ity

that

tw

o h

ybri

d gu

inea

pig

s w

ith

bla

ck h

air

colo

rin

g w

ill

prod

uce

a gu

inea

pig

wit

h w

hit

e h

air

colo

rin

g?25

%

Pra

ctic

e (

Ave

rag

e)

Sp

ecia

l Pro

du

cts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-8

8-8



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csS

pec

ial P

rod

uct

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-8

8-8

©G

lenc

oe/M

cGra

w-H

ill50

1G

lenc

oe A

lgeb

ra 1

Lesson 8-8

Pre-

Act

ivit

yW

hen

is

the

pro

du

ct o

f tw

o b

inom

ials

als

o a

bin

omia

l?

Rea

d th

e in

trod

uct

ion

to

Les

son

8-8

at

the

top

of p

age

458

in y

our

text

ook.

Wh

at i

s a

mea

nt

by t

he

term

tri

nom

ial

prod

uct

?

a th

ree-

term

po

lyn

om

ial a

nsw

er w

hen

mu

ltip

lyin

g p

oly

no

mia

ls

Rea

din

g t

he

Less

on

1.R

efer

to

the

Key

Con

cept

s bo

xes

on p

ages

458

,459

,an

d 46

0.

a.W

hen

mu

ltip

lyin

g tw

o bi

nom

ials

,th

ere

are

thre

e sp

ecia

l pr

odu

cts.

Wh

at a

re t

he

thre

esp

ecia

l pr

odu

cts

that

may

res

ult

wh

en m

ult

iply

ing

two

bin

omia

ls?

squ

are

of

a su

m,s

qu

are

of

a d

iffe

ren

ce,p

rod

uct

of

a su

m a

nd

ad

iffe

ren

ce

b.

Exp

lain

wh

at i

s m

ean

t by

th

e n

ame

of e

ach

spe

cial

pro

duct

.

squ

are

of

a su

m:

squ

arin

g t

he

sum

of

two

mo

no

mia

ls;

squ

are

of

ad

iffe

ren

ce:

squ

arin

g t

he

dif

fere

nce

of

two

mo

no

mia

ls;

pro

du

ct o

f a

sum

an

d a

dif

fere

nce

:th

e p

rod

uct

of

the

sum

an

d t

he

dif

fere

nce

of

the

sam

e tw

o t

erm

sc.

Use

th

e ex

ampl

es i

n t

he

Key

Con

cept

s bo

xes

to c

ompl

ete

the

tabl

e.

Sym

bo

lsP

rod

uct

Exa

mp

leP

rod

uct

Sq

uar

e o

f a

Su

m(a

�b

)2a

2�

2ab

�b

2(x

�7)

2x

2�

14x

�49

Sq

uar

e o

f a

Dif

fere

nce

(a�

b)2

a2

�2a

b�

b2

(x�

4)2

x2

�8x

�16

Pro

du

ct o

f a

Su

m

and

a D

iffe

ren

ce(a

�b

)(a

�b

)a

2�

b2

(x�

9)(x

�9)

x2

�81

2.W

hat

is

anot

her

ph

rase

th

at d

escr

ibes

th

e pr

odu

ct o

f th

e su

m a

nd

diff

eren

ce o

f tw

o te

rms?

dif

fere

nce

of

squ

ares

Hel

pin

g Y

ou

Rem

emb

er

3.E

xpla

in h

ow F

OIL

can

hel

p yo

u r

emem

ber

how

man

y te

rms

are

in t

he

spec

ial

prod

uct

sst

udi

ed i

n t

his

les

son

.F

or

the

squ

are

of

a su

m a

nd

th

e sq

uar

e o

f a

dif

fere

nce

,th

e in

ner

an

d o

ute

r p

rod

uct

s ar

e eq

ual

,so

th

ere

are

are

on

lyth

ree

term

s.F

or

the

pro

du

ct o

f th

e su

m a

nd

dif

fere

nce

of

two

ter

ms,

two

of

the

pro

du

cts

for

FO

IL a

re o

pp

osi

tes.

Th

at m

ean

s th

at t

he

fin

al p

rod

uct

has

on

ly t

wo

ter

ms.

©G

lenc

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w-H

ill50

2G

lenc

oe A

lgeb

ra 1

Su

ms

and

Diff

eren

ces

of

Cu

bes

R

ecal

l th

e fo

rmu

las

for

fin

din

g so

me

spec

ial

prod

uct

s:

Per

fect

-squ

are

trin

omia

ls:

(a�

b)2

�a2

�2a

b�

b2or

(a

�b)

2�

a2�

2ab

�b2

Dif

fere

nce

of

two

squ

ares

:(a

�b)

(a�

b) �

a2�

b2

A p

atte

rn a

lso

exis

ts f

or f

indi

ng

the

cube

of

a su

m (

a�

b)3 .

1.F

ind

the

prod

uct

of

(a�

b)(a

�b)

(a�

b).

a3

�3a

2 b�

3ab

2�

b3

2.U

se t

he

patt

ern

fro

m E

xerc

ise

1 to

eva

luat

e (x

�2)

3 .

x3

�6x

2�

12x

�8

3.B

ased

on

you

r an

swer

to

Exe

rcis

e 1,

pred

ict

the

patt

ern

for

th

e cu

be o

f a

diff

eren

ce (

a�

b)3 .

a3

�3a

2 b�

3ab

2�

b3

4.F

ind

the

prod

uct

of

(a�

b)(a

�b)

(a�

b) a

nd

com

pare

it

to y

our

answ

er f

or E

xerc

ise

3.

a3

�3a

2 b�

3ab

2�

b3

5.U

se t

he

patt

ern

fro

m E

xerc

ise

4 to

eva

luat

e (x

�4)

3 .

x3

�12

x2

�48

x�

64

Fin

d e

ach

pro

du

ct.

6.(x

�6)

37.

(x�

10)3

x3

�18

x2

�10

8x�

216

x3

�30

x2

�30

0x�

1000

8.(3

x�

y)3

9.(2

x�

y)3

27x

3�

27x

2 y�

9xy

2�

y3

8x3

�12

x2 y

�6x

y2

�y

3

10.(

4x�

3y)3

11.(

5x�

2)3

64x

3�

144x

2 y�

108x

y2

�27

y3

125x

3�

150x

2�

60x

�8

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

8-8

8-8



1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. A

C

C

B

A

D

D

C

D

A

C

39n � 1

B

B

D

D

C

A

A

B

D

C

A

C

D

C

B

A

A

C

B

D

D

C

C

A

B

Chapter 8 Assessment Answer Key Form 1 Form 2APage 503 Page 504 Page 505

(continued on the next page)


12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 37n � 1

C

A

D

B

A

A

D

D

B

C

A

D

C

D

B

B

D

C

B

B

7�2x � 2

C

A

C

C

A

B

C

A

D

An

swer

s

Chapter 8 Assessment Answer Key Form 2A (continued) Form 2BPage 506 Page 507 Page 508

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B: 21 terms

length is 20 ft;width is 12 ft

6

2�25

�

16n2 � 8n � 1

49a2 � 9b2

25c2 � 40c � 16

6s3 � 14s2 � 22s � 30

8x4 � 2y4

10h3k3 � 5h2k5 �20h3k4

�x2 � y

19m2 � 2mn � 6n2

�4n2 � 6ny � 13y2

�x5y � 3x3y3 � xy � 4

6

about 1.09 � 102 or 109times greater

1.0 � 105; 100,000

127,000

4.98 � 10�4

�xy5�

�4sr7

4�

p3q

8a3n6 � 4a18n6

w15y12

�18m4n7

3y8

Chapter 8 Assessment Answer Key Form 2CPage 509 Page 510


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B: 11 terms

length is 10 ft;width is 5 ft

–2

13

4c2 � 4c � 1

4k2 � 25r2

25y2 � 60y � 36

6n3 � n2 � 10n � 3

9r4 � 25s4

6x4y2 � 15x3y3 �24x4y4

a3 � 3b

8y2 � 4y � 9

4m2 � 6m � 1

x3 � 3x2y � xy2 � 3y3

5

2 � 107; 20,000,000

about 1.43 � 1010

or 14,300,000,000 in.

0.00743

1.2556 � 104

3r5s

�2db

7

5�

��m3

2�

10a4b8

w9z21

�6a3b8

5x7

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Chapter 8 Assessment Answer Key Form 2DPage 511 Page 512

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1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:�1�2�3�4 0 1 2 43

length is 16 cm;width is 12 cm

5

2a2 � 7a � 5

4

(9x2 � 6x � 1) ft2

25r4 � 9s4

8x3 � 2x2y � 19xy2 � 5y3

�13

�m2 � �161�m � 2

8y2 � 20y � 28

�n3 � 25n2 � 12n � 21

3x2 � x � 5y

10u2v � 7uv � 6uv2

10w2 � 3w � 4

6

7x6y3 � 4x5y2 � 2xy � 6

9.0 � 10�6; 0.000009

5.832 � 10�4;0.0005832

1.96783 � 105

��125

1a6

11b�

�m12

x8

12�

��6cd4

3�

45x�2

25r4s14

�1861�h12

3a7b10

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Chapter 8 Assessment Answer Key Form 3Page 513 Page 514

Chapter 8 Assessment Answer KeyPage 515, Open-Ended Assessment

Scoring Rubric


Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofmultiplying and dividing monomials, using scientificnotation, the degree of a polynomial, and adding,subtracting, and multiplying polynomials.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of multiplyingand dividing monomials, using scientific notation, thedegree of a polynomial, and adding, subtracting, andmultiplying polynomials.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofmultiplying and dividing monomials, using scientificnotation, the degree of a polynomial, and adding,subtracting, and multiplying polynomials.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Graphs are mostly accurate.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Graphs may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofmultiplying and dividing monomials, using scientificnotation, the degree of a polynomial, and adding,subtracting, and multiplying polynomials.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Graphs are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations

An

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Chapter 8 Assessment Answer Key Page 515, Open-Ended Assessment

Sample Answers

1a. Students should recognize that the v terms were not subtractedcorrectly. Since �5v � ( � v)� �5v � v � �6v, the right side of theequation should be 3u � 6v.

1b. Students should recognize that theProduct of a Power property was notused correctly to multiply x2y(3x3). Theright side of the equation should be3x5y � 4x2y.

1c. Students should recognize that thepattern for the Square of a Sum was notused correctly. The middle term2(3a)(5b) was omitted. Since(3a � 5b)2 � (3a)2 � 2(3a)(5b) � (5b)2,the right side of the equation should be9a2 � 30ab � 25b2.

2a. The number 23.4 � 108 is not inscientific notation. A number inscientific notation is of the form a � 10b where 1 � a � 10. The numberneeds to be adjusted to fit the form ofscientific notation.23.4 � 108 � 2.34 � 10 � 108

� 2.34 � 109

2b. Using scientific notation for division ofvery large and small numbers results inthe division of two numbers that arebetween one and ten. The correctdecimal place is found by using theQuotient of Powers property on twopowers of ten. Thus, 22,100,000 �0.00000013 � 2.21 � 107�1.3 � 10�7

� 12..321

�

�

101

�

07

7 � 21.2.31

� 1100�

7

7 � 1.7 � 1014

3a. Sample Answer: The binomial x � 1 hasdegree 1.

3b. Sample Answer: The trinomialx2 � x � 1 has degree 2, yet themonomial x3 has degree 3.

4a. �24aa�3

2�4� (2a5)4 � 16a20

4b. �24aa�3

2�4� (

(24aa�

3

2))4

4 � 21566aa�

1

8

2 � 16a20

4c. Sample answer: When simplifyingmonomials, the order of applying theQuotient of Powers property and Powerof a Quotient property does not matter.

5a.

3x2 � 7xy � 2y2

The length of the rectangular region is3x � y. The width of the region isx � 2y.The rectangular area is made upof 3 x2-areas, 7 xy-areas, and 2 y2-areas.When added together the area of therectangular region is exactly equal tothe product of the two binomials.

3x � y

x � 2y

x

x

x x y

yy

In addition to the scoring rubric found on page A31, the following sample answers may be used as guidance in evaluating open-ended assessment items.


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. the sum of theexponents of all thevariables

12. the greatest degreeof any term

13. a number written asa product of a factorgreater than orequal to 1 and lessthan 10 and a powerof 10

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Quiz (Lessons 8–3 and 8–4)

Page 517

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10.

1.

2.

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5.

Quiz (Lesson 8–7)

Page 518

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7.

8.

9.

10.(4a3 � 12a2 � 4a � 12)

units3

(4a2 � 4) units2

a2 � 9b2

81x2 � 49

m4 � 4m2n � 4n2

4x2 � 36y2

a2 � 18a � 81

16h2 � 24h � 9

2c3 � 11c2 � 22c � 15

4m2 � 7m � 2

2

�3c3 � 74c2 � 4c

6x3y � 15x2y2 � 21xy3

3a � 5b

3x2 � 3x � 4

�3 � 4x � x2

5x3y � x2y3 � 3xy4 � y4

�3x3 � 4x2 � 2x � 12

4

2

2.87 � 1010;28,700,000,000

1.6 � 10�7; 0.00000016

0.0064871

6.1 � 10�6

5.76 � 104

C

�nr2

9�

y5

66 or 46,65627y4w8

64c8d2

�125x12y6

�12m3n7

x20

2r8

h

d

b

j

a

i

f

g

c

e

Chapter 8 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 8–1 and 8–2) Quiz (Lessons 8–5 and 8–6)

Page 516 Page 517 Page 518

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1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14. 9x2 � 12x � 4

�13

�a2 � a � �35

�

1.38 � 109; 1,380,000,000

�3xy3

2�

elimination (x); (1, –1)

y

xO

x � y � 4

y � x

(2, 2)

one solution; (2, 2)

8x � 12y � 62

�y � y � 3�23

��{u � u � 3}

y � 4x � 9

$122.50 per yr.

f(x) � 2x � 2

c � b � �b1

�

10

8x4 � 2a2x3 �3a4x2 � 8x

4x3 � 2x2 � 3x � 6

4

�5m6

7y3r5�

�5x

29y�

7.553 � 10�4; 0.0007553

2.34 � 102; 234

A

D

C

A

A

A

B

Chapter 8 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 519 Page 520


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

16.

17. DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

. 0 5 7

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

5

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

3 . 5 0

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 0 8

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 8 Assessment Answer KeyStandardized Test Practice

Page 521 Page 522

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Documents

Chapter 8 Resource Masters - Morgan Park High School · 2010. 5. 3. · © Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide to Using the Chapter 8 Resource Masters The Fast