Chapter 8 Resource Masters - Math Class · ©Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide to Using the Chapter 8 Resource Masters The Fast File Chapter Resource system

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

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

Study Guide and InterventionScientific Notation

NAME ______________________________________________ DATE ____________ PERIOD _____

6-38-3


Less

on 8

-3

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

NAME ______________________________________________ DATE ____________ PERIOD _____

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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 # 1021.68 # 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 _____

8-38-3

Reading to Learn MathematicsScientific Notation

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3


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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 _____

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Reading to Learn MathematicsPolynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4


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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

NAME ______________________________________________ DATE ____________ PERIOD _____

6-58-5


Less

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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

ExampleExampleHorizontal 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)


Find each sum or difference.

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

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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 VolumesArea 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 yxx

r

h

r

h

r

1$3

Enrichment

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8-58-5

Study Guide and InterventionMultiplying a Polynomial by a Monomial

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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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ExercisesExercises

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-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

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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

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Study Guide and InterventionMultiplying Polynomials

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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

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8-78-7

ExampleExample

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Less

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-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

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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


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ExampleExample

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Less

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-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

NAME ______________________________________________ DATE ____________ PERIOD _____

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

© Glencoe/McGraw-Hill A2 Glencoe Algebra 1

Stu

dy

Gu

ide

and I

nte

rven

tion

Mul

tiply

ing

Mon

omia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

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6-1

8-1

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Lesson 8-1

Mul

tipl

y M

onom

ials

A m

onom

iali

s a

num

ber,

a va

riab

le,o

r a

prod

uct

of a

num

ber

and

one

or m

ore

vari

able

s.A

n ex

pres

sion

of

the

form

xn

is c

alle

d a

pow

eran

d re

pres

ents

the

prod

uct

you

obta

in w

hen

xis

use

d as

a f

acto

r n

tim

es.T

o m

ulti

ply

two

pow

ers

that

hav

eth

e sa

me

base

,add

the

exp

onen

ts.

Pro

duct

of

Pow

ers

For

any

num

ber

aan

d al

l int

eger

s m

and

n, a

m!

an"

am#

n.

Sim

pli

fy (

3x6 )

(5x2

).

(3x6

)(5x

2 ) "

(3)(

5)(x

6!

x2)

Ass

ocia

tive

Pro

pert

y

"(3

!5)

(x6

# 2

)P

rodu

ct o

f Pow

ers

"15

x8S

impl

ify.

The

pro

duct

is 1

5x8 .

Sim

pli

fy (

!4a

3 b)(

3a2 b

5 ).

($4a

3 b)(

3a2 b

5 )"

($4)

(3)(

a3!

a2)(

b!

b5)

"$

12(a

3 #

2)(

b1 #

5)

"$

12a5

b6

The

pro

duct

is $

12a5

b6.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sim

pli

fy.

1.y(

y5)

2.n2

!n7

3.($

7x2 )

(x4 )

y6

n9

!7x

6

4.x(

x2)(

x4)

5.m

!m

56.

($x3

)($

x4)

x7m

6x7

7.(2

a2)(

8a)

8.(r

s)(r

s3)(

s2)

9.(x

2 y)(

4xy3

)

16a3

r2s6

4x3 y

4

10.

(2a3

b)(6

b3)

11.(

$4x

3 )($

5x7 )

12.(

$3j

2 k4 )

(2jk

6 )

4a3 b

420

x10

!6j

3 k10

13.(

5a2 b

c3) !

abc4"

14.(

$5x

y)(4

x2)(

y4)

15.(

10x3

yz2 )

($2x

y5z)

a3b

2 c7

!20

x3 y

5!

20x

4 y6 z

3

1 % 5

1 % 3

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Pow

ers

of M

onom

ials

An

expr

essi

on o

f th

e fo

rm (x

m)n

is c

alle

d a

pow

er o

f a

pow

eran

d re

pres

ents

the

pro

duct

you

obt

ain

whe

n xm

is u

sed

as a

fac

tor

nti

mes

.To

find

the

pow

er o

f a

pow

er,m

ulti

ply

expo

nent

s.

Pow

er o

f a

Pow

erF

or a

ny n

umbe

r a

and

all i

nteg

ers

man

d n,

(am

)n"

amn .

Pow

er o

f a

Pro

duct

For

any

num

ber

aan

d al

l int

eger

s m

and

n, (

ab)m

"am

bm.

Sim

pli

fy (

!2a

b2)3

(a2 )

4 .

($2a

b2)3

(a2 )

4"

($2a

b2)3

(a8 )

Pow

er o

f a P

ower

"($

2)3 (

a3)(

b2)3

(a8 )

Pow

er o

f a P

rodu

ct

"($

2)3 (

a3)(

a8)(

b2)3

Com

mut

ativ

e P

rope

rty

"($

2)3 (

a11 )

(b2 )

3P

rodu

ct o

f Pow

ers

"$

8a11

b6P

ower

of a

Pow

er

The

pro

duct

is $

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

2 y4 )

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

12n1

2 y10

!24

3a15

n8

!76

8x14

y2

1 % 5Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Mul

tiply

ing

Mon

omia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 8-1)


Ans

wer

s

Skil

ls P

ract

ice

Mul

tiply

ing

Mon

omia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1

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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

umbe

r an

d an

exa

mpl

e of

a c

onst

ant.

2.a

$b

No;

This

is t

he d

iffer

ence

,not

the

pro

duct

,of

two

vari

able

s.

3.N

o;Th

is is

the

quo

tient

,not

the

pro

duct

,of

two

vari

able

s.

4.y

Yes;

Sin

gle

vari

able

s ar

e m

onom

ials

.

5.j3

kYe

s;Th

is is

the

pro

duct

of

two

vari

able

s.

6.2a

#3b

No;

This

is t

he s

um o

f tw

o m

onom

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)8c

5 d2

19.(

102 )

310

6or

1,0

00,0

0020

.(p3

)12

p36

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

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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;th

is in

volv

es t

he q

uotie

nt,n

ot t

he p

rodu

ct,o

f va

riab

les.

2.Ye

s;th

is is

the

pro

duct

of

a nu

mbe

r,,a

nd t

wo

vari

able

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)($

2g5 )

!8g

8 h

7.($

15xy

4 )!$

xy3 "

5x2 y

78.

($xy

)3(x

z)!

x4 y

3 z

9.($

18m

2 n)2!$

mn2

"!54

m5 n

410

.(0.

2a2 b

3 )2

0.04

a4b

6

11. !

p "2p

212

. !cd

3 "2c2

d6

13.(

0.4k

3 )3

0.06

4k9

14.[

(42 )

2 ]2

48or

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

four

ligh

t sw

itch

es c

an b

e se

t in

24

way

s.A

pan

el o

f fi

ve li

ght

swit

ches

can

set

in t

wic

e th

is m

any

way

s.In

how

man

y w

ays

can

five

ligh

t sw

itch

es

be s

et?

25or

32

22.H

OB

BIE

STa

wa

wan

ts t

o in

crea

se h

er r

ock

colle

ctio

n by

a p

ower

of

thre

e th

is y

ear

and

then

incr

ease

it a

gain

by

a po

wer

of

two

next

yea

r.If

she

has

2 r

ocks

now

,how

man

yro

cks

will

she

hav

e af

ter

the

seco

nd y

ear?

26or

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 (A

vera

ge)

Mul

tiply

ing

Mon

omia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1



Rea

din

g t

o L

earn

Math

emati

csM

ultip

lyin

g M

onom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1

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Lesson 8-1

Pre-

Act

ivit

yW

hy

doe

s d

oubl

ing

spee

d q

uad

rup

le b

rak

ing

dis

tan

ce?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

1 at

the

top

of

page

410

in y

our

text

book

.

Fin

d tw

o ex

ampl

es in

the

tab

le t

o ve

rify

the

sta

tem

ent

that

whe

n sp

eed

isdo

uble

d,th

e br

akin

g di

stan

ce is

qua

drup

led.

Wri

te y

our

exam

ples

in t

heta

ble.

Spe

edB

raki

ng D

ista

nce

Spe

ed D

oubl

ed

Bra

king

Dis

tanc

e (m

iles

per

hour

)(f

eet)

(mile

s pe

r ho

ur)

Qua

drup

led

(fee

t)

2020

4080

3045

6018

0

Read

ing

the

Less

on

1.D

escr

ibe

the

expr

essi

on 3

xyus

ing

the

term

s m

onom

ial,

cons

tant

,var

iabl

e,an

d pr

oduc

t.

The

mon

omia

l 3xy

is t

he p

rodu

ct o

f th

e co

nsta

nt 3

and

the

var

iabl

es x

and

y.

2.C

ompl

ete

the

char

t by

cho

osin

g th

e pr

oper

ty t

hat

can

be u

sed

to s

impl

ify

each

expr

essi

on.T

hen

sim

plif

y th

e ex

pres

sion

.

Exp

ress

ion

Pro

pert

yE

xpre

ssio

n S

impl

ified

Pro

duct

of P

ower

s

35!

32P

ower

of a

Pow

er37

or 2

187

Pow

er o

f a P

rodu

ct

Pro

duct

of P

ower

s

(a3 )

4P

ower

of a

Pow

era1

2

Pow

er o

f a P

rodu

ct

Pro

duct

of P

ower

s

($4x

y)5

Pow

er o

f a P

ower

$

1024

x5 y

5

Pow

er o

f a P

rodu

ct

Hel

ping

You

Rem

embe

r

3.W

rite

an

exam

ple

of e

ach

of t

he t

hree

pro

pert

ies

of p

ower

s di

scus

sed

in t

his

less

on.

The

n,us

ing

the

exam

ples

,exp

lain

how

the

pro

pert

y is

use

d to

sim

plif

y th

em.

Sam

ple

answ

er:F

or z

2$

z5 ,

sinc

e th

e ba

ses

are

the

sam

e,us

e th

eP

rodu

ct o

f P

ower

s P

rope

rty

and

add

the

expo

nent

s to

get

z7 .

For

(a4 )

3 ,us

e th

e P

ower

of

a P

ower

Pro

pert

y.M

ultip

ly t

he e

xpon

ents

to

get

a12 .

For

(3rs

)3,u

se t

he P

ower

of

a P

rodu

ct P

rope

rty.

Rai

se t

he c

onst

ant

and

each

var

iabl

e to

the

pow

er t

o ge

t 27

r3s

3 .

©G

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0G

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An

Wan

g A

n W

ang

(192

0–19

90)

was

an

Asi

an-A

mer

ican

who

bec

ame

one

of t

hepi

onee

rs o

f th

e co

mpu

ter

indu

stry

in t

he U

nite

d St

ates

.He

grew

up

inSh

angh

ai,C

hina

,but

cam

e to

the

Uni

ted

Stat

es t

o fu

rthe

r hi

s st

udie

sin

sci

ence

.In

1948

,he

inve

nted

a m

agne

tic

puls

e co

ntro

lling

dev

ice

that

vas

tly

incr

ease

d th

e st

orag

e ca

paci

ty o

f co

mpu

ters

.He

late

rfo

unde

d hi

s ow

n co

mpa

ny,W

ang

Lab

orat

orie

s,an

d be

cam

e a

lead

er in

the

deve

lopm

ent

of d

eskt

op c

alcu

lato

rs a

nd w

ord

proc

essi

ng s

yste

ms.

In 1

988,

Wan

g w

as e

lect

ed t

o th

e N

atio

nal I

nven

tors

Hal

l of

Fam

e.

Dig

ital

com

pute

rs s

tore

info

rmat

ion

as n

umbe

rs.B

ecau

se t

heel

ectr

onic

cir

cuit

s of

a c

ompu

ter

can

exis

t in

onl

y on

e of

tw

o st

ates

,op

en o

r cl

osed

,the

num

bers

tha

t ar

e st

ored

can

con

sist

of

only

tw

odi

gits

,0 o

r 1.

Num

bers

wri

tten

usi

ng o

nly

thes

e tw

o di

gits

are

cal

led

bin

ary

nu

mbe

rs.T

o fi

nd t

he d

ecim

al v

alue

of a

bin

ary

num

ber,

you

use

the

digi

ts t

o w

rite

a p

olyn

omia

l in

2.F

or in

stan

ce,t

his

is h

ow t

ofi

nd t

he d

ecim

al v

alue

of

the

num

ber

1001

101 2

.(T

he s

ubsc

ript

2

indi

cate

s th

at t

his

is a

bin

ary

num

ber.)

1001

101 2

"1

&2 %

6 %#

0 &

2 %5 %

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Fin

d t

he

dec

imal

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ue

of e

ach

bin

ary

nu

mbe

r.

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1000

0 216

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195

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Wri

te e

ach

dec

imal

nu

mbe

r as

a b

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y n

um

ber.

5.8

1000

6.11

1011

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117

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101

9.T

he c

hart

at

the

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t sh

ows

a se

t of

dec

imal

co

de n

umbe

rs t

hat

is u

sed

wid

ely

in s

tori

ng

lett

ers

of t

he a

lpha

bet

in a

com

pute

r's

mem

ory.

Fin

d th

e co

de n

umbe

rs fo

r th

e le

tter

s of

you

r na

me.

The

n w

rite

the

cod

e fo

r yo

ur n

ame

usin

g bi

nary

num

bers

. Ans

wer

s w

ill v

ary.

The

Am

eric

an S

tand

ard

Gui

de fo

r In

form

atio

n In

terc

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e (A

SC

II)

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rich

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NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1



Ans

wer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Div

idin

g M

onom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-2

8-2

©G

lenc

oe/M

cGra

w-H

ill46

1G

lenc

oe A

lgeb

ra 1

Lesson 8-2

Quo

tien

ts o

f M

onom

ials

To d

ivid

e tw

o po

wer

s w

ith

the

sam

e ba

se,s

ubtr

act

the

expo

nent

s.

Quo

tient

of

Pow

ers

For

all

inte

gers

man

d n

and

any

nonz

ero

num

ber

a,

"am

$n .

Pow

er o

f a

Quo

tient

For

any

inte

ger

man

d an

y re

al n

umbe

rs a

and

b, b

'0,

!"m

".

am% bm

a % b

am% an

Sim

pli

fy

.Ass

um

e

nei

ther

an

or b

is e

qual

to

zero

.

"!

"!"

Gro

up p

ower

s w

ith th

e sa

me

base

.

"(a

4 $

1 )(b

7 $

2 )Q

uotie

nt o

f Pow

ers

"a3

b5S

impl

ify.

The

quo

tien

t is

a3 b

5.

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

qual

to

zero

.

!"3

"P

ower

of a

Quo

tient

"P

ower

of a

Pro

duct

"P

ower

of a

Pow

er

"Q

uotie

nt o

f Pow

ers

The

quo

tien

t is

.

8a9 b

9%

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

2a3 b

5#

3b2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o ze

ro.

1.53

or 1

252.

m2

3.p

3 n3

4.a

5.y

6.!

y2

7.y

28.

!"3

8a3 b

39.

!"3

p6 q

6

10. !

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v4

11. !

"4r4

s812

.r4

s4

r7 s7 t

2% s3 r

3 t2

81 # 163r

6 s3

% 2r5 s

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3% v4 w

3

64 # 274p

4 q4

% 3p2 q

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axy

6% y4 x

1 # 7$

2y7

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% x5 y2

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% p2 nm

6% m

455% 52

©G

lenc

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cGra

w-H

ill46

2G

lenc

oe A

lgeb

ra 1

Neg

ativ

e Ex

pone

nts

Any

non

zero

num

ber

rais

ed t

o th

e ze

ro p

ower

is 1

;for

exa

mpl

e,($

0.5)

0"

1.A

ny n

onze

ro n

umbe

r ra

ised

to

a ne

gati

ve p

ower

is e

qual

to

the

reci

proc

al o

f th

e nu

mbe

r ra

ised

to

the

oppo

site

pow

er;f

or e

xam

ple,

6$3

".T

hese

def

init

ions

can

be

used

to

sim

plif

y ex

pres

sion

s th

at h

ave

nega

tive

exp

onen

ts.

Zero

Exp

onen

tF

or a

ny n

onze

ro n

umbe

r a,

a0

"1.

Neg

ativ

e E

xpon

ent

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pert

yF

or a

ny n

onze

ro n

umbe

r a

and

any

inte

ger

n, a

$n

"an

d "

an.

The

sim

plif

ied

form

of

an e

xpre

ssio

n co

ntai

ning

neg

ativ

e ex

pone

nts

mus

t co

ntai

n on

lypo

siti

ve e

xpon

ents

.

Sim

pli

fy

.Ass

um

e th

at t

he

den

omin

ator

is

not

equ

al t

o ze

ro.

"!

"!"!

"!"

Gro

up p

ower

s w

ith th

e sa

me

base

.

"(a

$3

$2 )

(b6

$6 )

(c5 )

Quo

tient

of P

ower

s an

d N

egat

ive

Exp

onen

t Pro

pert

ies

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5 b0 c

5S

impl

ify.

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"(1)c

5N

egat

ive

Exp

onen

t and

Zer

o E

xpon

ent P

rope

rtie

s

"S

impl

ify.

The

sol

utio

n is

.

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o ze

ro.

1.25

or 3

22.

m5

3.

4.b

5.6.

a6b

8

7.x

68.

9.

10.

11. !

"01

12.

!m

3# 32

n10($

2mn2 )

$3

%%

4m$

6 n4

4m2 n

2% 8m

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3 t$

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# u11(3

st)2 u

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%%

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2 u7

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a$1 b

)2%

%(b

2 )4

x4 y0

% x$2

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3 )2

% (ab)

$2

w # 4y2

($x$

1 y)0

%%

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1 y2

b$4

% b$5

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p$8

% p3m

% m$

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3

c5% 4a

5

c5% 4a

51 % a51 % 41 % 41 % 4

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4a$

3 b6

%%

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6 c$

5

4a$

3 b6

##

16a

2 b6 c

$5

1% a$

n1 % an

1 % 63

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Div

idin

g M

onom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Div

idin

g M

onom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2

©G

lenc

oe/M

cGra

w-H

ill46

3G

lenc

oe A

lgeb

ra 1

Lesson 8-2

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o ze

ro.

1.61

or 6

2.94

or 6

561

3.x

24.

5.6.

3d

7.8.

u2

9.a2

b3

10.

m4

11.

!12

.!

4x2 y

z3

13. !

"214

.4$

4

15.8

$2

16. !

"$2

17. !

"$1

18.

h9

19.k

0 (k4

)(k$

6 )20

.k$

1 (!$

6 )(m

3 )

21.

22. !

"01

23.

24.

3x5 y

2

25.

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.!

8x5 y

2#

z48

x6 y7 z

5%

%$

6xy5 z

63 # u

4$

15w

0 u$

1%

%5u

3

15x6 y

$9

% 5xy$

11g

4 h2

#f5

f$5 g

4% h$

2

16p5 q

2% 2p

3 q3

1 # f11f$

7% f4

m3

# k!6

1 # k2

h3% h$

611 # 9

9 % 11

9 # 255 % 3

1 # 64

1# 25

616

p14# 49

s44p

7% 7s

2

32x3 y

2 z5

%%

$8x

yz2

3w # u3

$21

w5 u

2%

%7w

4 u5

m7 n

2% m

3 n2

a3 b5

% ab2

w4 u

3% w

4 un

4# 3

12n5

% 36n

9d7

% 3d6

1 # m2

m % m3

1 # s2r3 s

2% r3 s

4x4% x2

912% 98

65% 64

©G

lenc

oe/M

cGra

w-H

ill46

4G

lenc

oe A

lgeb

ra 1

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o ze

ro.

1.84

or 4

096

2.a3

b3

3.y

4.m

n5.

!6.

2yz

7.!

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!"2

9.!

10.x

3 (y$

5 )(x

$8 )

11.p

(q$

2 )(r

$3 )

12.1

2$2

13. !

"$2

14. !

"$4

15.

2rs5

16.

!17

.2c

4 f7

18. !

"01

19.

20.

!21

.

22.

23.

24.

25. !

"$5

26. !

"$1

27. !

"$2

28.B

IOLO

GY

A la

b te

chni

cian

dra

ws

a sa

mpl

e of

blo

od.A

cub

ic m

illim

eter

of

the

bloo

dco

ntai

ns 2

23w

hite

blo

od c

ells

and

225

red

bloo

d ce

lls.W

hat

is t

he r

atio

of

whi

te b

lood

cells

to

red

bloo

d ce

lls?

29. C

OU

NTI

NG

The

num

ber

of t

hree

-let

ter

“wor

ds”

that

can

be

form

ed w

ith

the

Eng

lish

alph

abet

is 2

63.T

he n

umbe

r of

five

-let

ter

“wor

ds”

that

can

be

form

ed is

265

.How

man

yti

mes

mor

e fi

ve-l

ette

r “w

ords

”ca

n be

for

med

tha

n th

ree-

lett

er “

wor

ds”?

676

1# 48

4

9x2

# 4y2 z

62x

3 y2 z

% 3x4 y

z$2

c8# 7d

2 e4

7c$

3 d3

% c5 de$

4q10# r25

q$1 r

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b7

(2a$

2 b)$

3%

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2 b4

j# k15

(j$

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$4

%j3 k

3m

2# n

2m

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$5

%%

(m4 n

3 )$

1

r # 27r4

% (3r)

36t

2 u4

#v9

$12

t$1 u

5 v$

4%

%2t

$3 u

v5g

8 h2

#9

6f$

2 g3 h

5%

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f$2 g

$5 h

3

x$3 y

5%4$

38c

3 d2 f

4%

%4c

$1 d

2 f$

33 # u4

$15

w0 u

$1

%%

5u3

22r3 s

2% 11

r2 s$

381 # 25

64 % 3

49 # 93 % 7

1# 14

4p

# q2 r

31

# x5 y5

1 # 6c3

$4c

2% 24

c536

w10

# 49p12

s66w

5% 7p

6 s3

64f9 g

3# 27

h184f

3 g% 3h

6

8y7 z

6% 4y

6 z5

5d2

#4

5c2 d

3% $

4c2 d

m5 n

p% m

4 p

xy2

% xya4 b

6% ab

388% 84

Pra

ctic

e (A

vera

ge)

Div

idin

g M

onom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2



Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csD

ivid

ing

Mon

omia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2

©G

lenc

oe/M

cGra

w-H

ill46

5G

lenc

oe A

lgeb

ra 1

Lesson 8-2

Pre-

Act

ivit

yH

ow c

an y

ou c

omp

are

pH

lev

els?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

2 at

the

top

of

page

417

in y

our

text

book

.

•In

the

for

mul

a c

"!

"pH,i

dent

ify

the

base

and

the

exp

onen

t.

base

%,e

xpon

ent

%pH

•H

ow d

o yo

u th

ink

cw

ill c

hang

e as

the

exp

onen

t in

crea

ses?

cw

ill d

ecre

ase.

Read

ing

the

Less

on

1.E

xpla

in w

hat

the

stat

emen

t "

am$

nm

eans

.

To d

ivid

e tw

o po

wer

s th

at h

ave

the

sam

e ba

se,s

ubtr

act

the

expo

nent

s.

2.To

find

cin

the

form

ula

c"

!"pH

,you

can

find

the

pow

er o

f the

num

erat

or,t

he p

ower

of

the

deno

min

ator

,and

div

ide.

Thi

s is

an

exam

ple

of w

hat

prop

erty

?P

ower

of

a Q

uotie

nt P

rope

rty

3.U

se t

he Q

uoti

ent

of P

ower

s P

rope

rty

to e

xpla

in w

hy 3

0"

1.S

ampl

e an

swer

:

%1.

The

Quo

tient

of

Pow

ers

Pro

pert

y sa

ys t

hat

whe

n yo

u di

vide

two

pow

ers

that

hav

e th

e sa

me

base

,you

sub

trac

t th

e ex

pone

nts.

So

%30

.

4.C

onsi

der

the

expr

essi

on 4

$3 .

a.E

xpla

in w

hy t

he e

xpre

ssio

n 4$

3is

not

sim

plif

ied.

An

expr

essi

on in

volv

ing

expo

nent

s is

not

con

side

red

sim

plifi

ed if

the

exp

ress

ion

cont

ains

nega

tive

expo

nent

s.b.

Def

ine

the

term

rec

ipro

cal.

The

reci

proc

al o

f a

num

ber

is 1

div

ided

by

the

num

ber.

c.4$

3is

the

rec

ipro

cal o

f w

hat

pow

er o

f 4?

43

d.W

hat

is t

he s

impl

ifie

d fo

rm o

f 4$

3 ?or

Hel

ping

You

Rem

embe

r

5.D

escr

ibe

how

you

wou

ld h

elp

a fr

iend

who

nee

ds t

o si

mpl

ify

the

expr

essi

on

.

Div

ide

the

cons

tant

s an

d gr

oup

pow

ers

with

the

sam

e ba

se t

o ge

t

!"!

".Use

the

Quo

tient

of

Pow

ers

Pro

pert

y to

get

(2)

(x2!

5 ) o

r (2

)(x!

3 ).

To s

impl

ify (

2)( x

!3 )

,use

the

Neg

ativ

e E

xpon

ent

Pro

pert

y to

get

(2) !

",or

.2 # x3

1 # x3x2# x5

4 # 2

4x2

% 2x5

1 # 641 # 43

34# 34

34# 34

1 % 10am % an

1 # 10

1 % 10

©G

lenc

oe/M

cGra

w-H

ill46

6G

lenc

oe A

lgeb

ra 1

Patt

erns

with

Pow

ers

Use

you

r ca

lcul

ator

,if

nece

ssar

y,to

com

plet

e ea

ch p

atte

rn.

a.21

0"

b.51

0"

c.41

0"

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"

Stu

dy

the

pat

tern

s fo

r a,

b,an

d c

abo

ve.T

hen

an

swer

th

e qu

esti

ons.

1.D

escr

ibe

the

patt

ern

of t

he e

xpon

ents

fro

m t

he t

op o

f ea

ch c

olum

n to

the

bot

tom

.Th

e ex

pone

nts

decr

ease

by

one

from

eac

h ro

w t

o th

e on

e be

low

.

2.D

escr

ibe

the

patt

ern

of t

he p

ower

s fr

om t

he t

op o

f th

e co

lum

n to

the

bot

tom

.To

get

each

pow

er,d

ivid

e th

e po

wer

on

the

row

abo

ve b

y th

e ba

se (

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

rul

e.Te

st it

on

patt

erns

tha

t yo

u ob

tain

usi

ng 2

2,25

,and

24

as b

ases

.A

ny n

onze

ro n

umbe

r to

the

zer

o po

wer

equ

als

one.

Stu

dy

the

pat

tern

bel

ow.T

hen

an

swer

th

e qu

esti

ons.

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 d

o 0$

1 ,0$

2 ,an

d 0$

3no

t ex

ist?

Neg

ativ

e ex

pone

nts

are

not

defin

ed u

nles

s th

e ba

se is

non

zero

.

6.B

ased

upo

n th

e pa

tter

n,ca

n yo

u de

term

ine

whe

ther

00

exis

ts?

No,

sinc

e th

e pa

tter

n 0n

%0

brea

ks d

own

for

n&

1.

7.T

he s

ymbo

l 00

is c

alle

d an

in

dete

rmin

ate,

whi

ch m

eans

tha

t it

has

no

uniq

ue v

alue

.T

hus

it d

oes

not

exis

t as

a u

niqu

e re

al n

umbe

r.W

hy d

o yo

u th

ink

that

00

cann

ot e

qual

1?

Ans

wer

s w

ill v

ary.

One

ans

wer

is t

hat

if 00

%1,

then

1 %

%%

!"0 ,

whi

ch is

a f

alse

res

ult,

sinc

e di

visi

on b

y ze

ro is

not

allo

wed

.Thu

s,00

cann

ot e

qual

1.

1 # 010# 00

1 # 1

?

45

216

254

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58

256

625

1610

2431

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128

65,5

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626

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512

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765,

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1024

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rich

men

t

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____

____

____

____

____

____

____

____

____

____

____

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____

____

____

PE

RIO

D__

___

8-2

8-2



Stu

dy

Gu

ide

and I

nte

rven

tion

Sci

entif

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otat

ion

NA

ME

____

____

____

____

____

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6-3

8-3

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7G

lenc

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lgeb

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Lesson 8-3

Scie

ntif

ic N

otat

ion

Kee

ping

tra

ck o

f pla

ce v

alue

in v

ery

larg

e or

ver

y sm

all n

umbe

rsw

ritt

en in

sta

ndar

d fo

rm m

ay b

e di

ffic

ult.

It is

mor

e ef

fici

ent

to w

rite

suc

h nu

mbe

rs in

scie

ntif

ic n

otat

ion.

A n

umbe

r is

exp

ress

ed in

sci

enti

fic

nota

tion

whe

n it

is w

ritt

en a

s a

prod

uct

of t

wo

fact

ors,

one

fact

or t

hat

is g

reat

er t

han

or e

qual

to

1 an

d le

ss t

han

10 a

nd o

nefa

ctor

tha

t is

a p

ower

of

ten.

Sci

entif

ic N

otat

ion

Anu

mbe

r is

in s

cien

tific

not

atio

n w

hen

it is

in th

e fo

rm a

&10

n , w

here

1 (

a)

10

and

nis

an

inte

ger.

Exp

ress

3.5

2 '

104

inst

and

ard

not

atio

n.

3.52

&10

4"

3.52

&10

,000

"35

,200

The

dec

imal

poi

nt m

oved

4 p

lace

s to

the

righ

t.

Exp

ress

37,

600,

000

insc

ien

tifi

c n

otat

ion

.

37,6

00,0

00 "

3.76

&10

7

The

dec

imal

poi

nt m

oved

7 p

lace

s so

tha

t it

is b

etw

een

the

3 an

d th

e 7.

Sinc

e 37

,600

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*1,

the

expo

nent

is p

osit

ive.

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6.2

1 '

10!

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stan

dar

d n

otat

ion

.

6.21

&10

$5

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21 &

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21 &

0.00

001

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dec

imal

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nt m

oved

5 p

lace

s to

the

left

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0.0

0005

49 i

nsc

ien

tifi

c n

otat

ion

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0.00

0054

9 "

5.49

&10

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dec

imal

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nt m

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t it

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e 0.

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Exam

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3Ex

ampl

e 3

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ampl

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Exer

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Exp

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h n

um

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in s

tan

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d n

otat

ion

.

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65 &

105

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02 &

10$

43.

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3 &

108

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2 '

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lenc

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Prod

ucts

and

Quo

tien

ts w

ith

Scie

ntif

ic N

otat

ion

You

can

use

prop

erti

es o

fpo

wer

s to

com

pute

wit

h nu

mbe

rs w

ritt

en in

sci

enti

fic

nota

tion

.

Eva

luat

e (6

.7 '

103 )

(2 '

10!

5 ).E

xpre

ss t

he

resu

lt i

n s

cien

tifi

c an

dst

and

ard

not

atio

n.

(6.7

&10

3 )(2

&10

$5 )

"(6

.7 &

2)(1

03&

10$

5 )A

ssoc

iativ

e P

rope

rty

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10$

2P

rodu

ct o

f Pow

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"(1

.34

&10

1 ) &

10$

213

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1.34

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(101

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Ass

ocia

tive

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pert

y

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34 &

10$

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0.1

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rodu

ct o

f Pow

ers

The

sol

utio

n is

1.3

4 &

10$

1or

0.1

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Eva

luat

e .E

xpre

ss t

he

resu

lt i

n s

cien

tifi

c an

d

stan

dar

d n

otat

ion

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"!

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ocia

tive

Pro

pert

y

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368

&10

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uotie

nt o

f Pow

ers

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$1 )

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30.

368

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10$

1

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68 &

(10$

1&

103 )

Ass

ocia

tive

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pert

y

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68 &

102

or 3

68P

rodu

ct o

f Pow

ers

The

sol

utio

n is

3.6

8 &

102

or 3

68.

Eva

luat

e.E

xpre

ss e

ach

res

ult

in

sci

enti

fic

and

sta

nd

ard

not

atio

n.

1.2.

3.(3

.2 &

10$

2 )(2

.0 &

102 )

7 '

10;7

01.

5 '

103 ;

1500

6.4

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0 ;6.

4

4.5.

(7.7

&10

5 )(2

.1 &

102 )

6.

5.28

'10

3 ;52

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8 ;16

1,70

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2 ;0.

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105 )

(1.5

&10

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1 ;49

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2 ;30

01.

6 '

10!

6 ;0.

0000

016

10.F

UEL

CO

NSU

MPT

ION

Nor

th A

mer

ica

burn

ed 4

.5 &

1016

BT

U o

f pe

trol

eum

in 1

998.

At

this

rat

e,ho

w m

any

BT

U’s

will

be

burn

ed in

9 y

ears

?So

urce

: The

New

York

Tim

es 2

001

Alm

anac

4.05

'10

17

11.O

IL P

RO

DU

CTI

ON

If t

he U

nite

d St

ates

pro

duce

d 6.

25 &

109

barr

els

of c

rude

oil

in19

98,a

nd C

anad

a pr

oduc

ed 1

.98

&10

9ba

rrel

s,w

hat

is t

he q

uoti

ent

of t

heir

pro

duct

ion

rate

s? W

rite

a s

tate

men

t us

ing

this

quo

tien

t.So

urce

: The

New

York

Tim

es 2

001

Alm

anac

Abo

ut 3

.16;

Sam

ple

answ

er:T

he U

nite

d S

tate

s pr

oduc

es m

ore

than

3

times

the

cru

de o

il of

Can

ada.

4 &

10$

4%

%2.

5 &

102

3.3

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%%

1.1

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8%

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2 &

1010

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72 &

10$

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4 &

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12

3 &

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12%

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1.4

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4%

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&10

2

108

% 105

1.50

88%

4.1

1.50

88 &

108

%%

4.1

&10

5

1.50

88 '

108

##

4.1

'10

5

Stu

dy

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ide

and I

nte

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(c

onti

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)

Sci

entif

ic N

otat

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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ATE

____

____

____

PE

RIO

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8-3

8-3

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises



Ans

wer

s

Skil

ls P

ract

ice

Sci

entif

ic N

otat

ion

NA

ME

____

____

____

____

____

____

____

____

____

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____

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ATE

____

____

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PE

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8-3

8-3

©G

lenc

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9G

lenc

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ra 1

Lesson 8-3

Exp

ress

eac

h n

um

ber

in s

tan

dar

d n

otat

ion

.

1.4

&10

32.

2 &

108

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2 &

105

4000

200,

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000

320,

000

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5.9

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$2

6.4.

7 &

10$

7

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0.09

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47

AST

RO

NO

MY

Exp

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th

e n

um

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in e

ach

sta

tem

ent

in s

tan

dar

d n

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.

7.T

he d

iam

eter

of

Jupi

ter

is 1

.429

84 &

105

kilo

met

ers.

142,

984

8.T

he s

urfa

ce d

ensi

ty o

f th

e m

ain

ring

aro

und

Jupi

ter

is 5

&10

$6

gram

s pe

r ce

ntim

eter

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0.00

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9.T

he m

inim

um d

ista

nce

from

Mar

s to

Ear

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5.4

5 &

107

kilo

met

ers.

54,5

00,0

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um

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in s

cien

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011

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5 '

104

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luat

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in

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enti

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and

sta

nd

ard

not

atio

n.

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&10

7 )(2

&10

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5 &

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2 )(1

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4 )

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108 ;

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xpre

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he

nu

mbe

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eac

h s

tate

men

t in

sta

nd

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not

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7.A

n el

ectr

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as a

neg

ativ

e ch

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of 1

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10$

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and

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31.B

IOLO

GY

A c

ubic

mill

imet

er o

f hu

man

blo

od c

onta

ins

abou

t 5

&10

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d bl

ood

cells

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adul

t hu

man

bod

y m

ay c

onta

in a

bout

5 &

106

cubi

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eter

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blo

od.A

bout

how

man

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d bl

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cells

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s su

ch a

hum

an b

ody

cont

ain?

abou

t 2.

5 '

1013

or 2

5 tr

illio

n

32.P

OPU

LATI

ON

The

pop

ulat

ion

of A

rizo

na is

abo

ut 4

.778

&10

6pe

ople

.The

land

are

a is

abou

t 1.

14 &

105

squa

re m

iles.

Wha

t is

the

pop

ulat

ion

dens

ity

per

squa

re m

ile?

abou

t 42

peo

ple

per

squa

re m

ile

2.01

5 &

10$

3%

%3.

1 &

102

1.68

&10

4%

%8.

4 &

10$

41.

82 &

105

%%

9.1

&10

7

1.17

&10

2%

%5

&10

$1

6.72

&10

3%

%4.

2 &

108

3.12

&10

3%

%1.

56 &

10$

3

Pra

ctic

e (A

vera

ge)

Sci

entif

ic N

otat

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-3

8-3



Rea

din

g t

o L

earn

Math

emati

csS

cien

tific

Not

atio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill47

1G

lenc

oe A

lgeb

ra 1

Lesson 8-3

Pre-

Act

ivit

yW

hy

is s

cien

tifi

c n

otat

ion

im

por

tan

t in

ast

ron

omy?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

3 at

the

top

of

page

425

in y

our

text

book

.

In t

he t

able

,eac

h m

ass

is w

ritt

en a

s th

e pr

oduc

t of

a n

umbe

r an

d a

pow

erof

10.

Loo

k at

the

fir

st f

acto

r in

eac

h pr

oduc

t.H

ow a

re t

hese

fac

tors

alik

e?

They

are

all

grea

ter

than

1 a

nd le

ss t

han

10.

Read

ing

the

Less

on

1.Is

the

num

ber

0.05

43 &

104

in s

cien

tifi

c no

tati

on?

Exp

lain

.

No;

the

first

fac

tor

is le

ss t

han

1.

2.C

ompl

ete

each

sen

tenc

e to

cha

nge

from

sci

enti

fic

nota

tion

to

stan

dard

not

atio

n.

a.To

exp

ress

3.6

4 &

106

in s

tand

ard

nota

tion

,mov

e th

e de

cim

al p

oint

plac

es t

o th

e .

b.To

exp

ress

7.8

25 &

10$

3in

sta

ndar

d no

tati

on,m

ove

the

deci

mal

poi

nt

plac

es t

o th

e .

3.C

ompl

ete

each

sen

tenc

e to

cha

nge

from

sta

ndar

d no

tati

on t

o sc

ient

ific

not

atio

n.

a.To

exp

ress

0.0

0078

65 in

sci

enti

fic

nota

tion

,mov

e th

e de

cim

al p

oint

pl

aces

to t

he r

ight

and

wri

te

.

b.To

exp

ress

54,

000,

000,

000

in s

cien

tifi

c no

tati

on,m

ove

the

deci

mal

poi

nt

plac

es t

o th

e le

ft a

nd w

rite

.

4.W

rite

pos

itiv

eor

neg

ativ

eto

com

plet

e ea

ch s

ente

nce.

a.po

wer

s of

10

are

used

to

expr

ess

very

larg

e nu

mbe

rs in

scie

ntif

ic n

otat

ion.

b.po

wer

s of

10

are

used

to

expr

ess

very

sm

all n

umbe

rs in

scie

ntif

ic n

otat

ion.

Hel

ping

You

Rem

embe

r

5.D

escr

ibe

the

met

hod

you

wou

ld u

se t

o es

tim

ate

how

man

y ti

mes

gre

ater

the

mas

s of

Satu

rn is

tha

n th

e m

ass

of P

luto

.

Div

ide

5.69

'10

26by

1.2

7 '

1022

.Sin

ce 5

.69

(1.

27 #

4.48

and

10

26(

1022

is 1

04,t

he m

ass

of M

ars

is a

bout

4.4

8 '

104

times

the

m

ass

of P

luto

.

Neg

ativ

e

Pos

itive

5.4

'10

1010

7.86

5 '

10!

44

left

3ri

ght

6

©G

lenc

oe/M

cGra

w-H

ill47

2G

lenc

oe A

lgeb

ra 1

Con

vert

ing

Met

ric

Uni

tsSc

ient

ific

not

atio

n is

con

veni

ent

to u

se f

or u

nit

conv

ersi

ons

in t

he m

etri

c sy

stem

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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

num

ber

ofm

eter

s (1

000)

in o

ne k

ilom

eter

.Exp

ress

both

num

bers

in s

cien

tifi

c no

tati

on.

"4.

3 &

103

The

ans

wer

is 4

.3 3

103

km

.

4.3

&10

6%

%1

&10

3

Con

vert

370

0 gr

ams

into

mil

ligr

ams.

Mul

tipl

y by

the

num

ber

of m

illig

ram

s(1

000)

in 1

gra

m.

(3.7

&10

3 )(1

&10

3 ) "

3.7

&10

6

The

re a

re 3

.7 &

106

mg

in 3

700

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 p

lane

t M

ars

has

a di

amet

er o

f 6.

76 &

103

km.W

hat

is t

hedi

amet

er o

f M

ars

in m

eter

s? E

xpre

ss t

he a

nsw

er in

bot

h sc

ient

ific

and

deci

mal

not

atio

n.6,

760,

000

m;6

.76

'10

6m

12.T

he d

ista

nce

from

ear

th t

o th

e su

n is

149

,590

,000

km

.Lig

httr

avel

s 3.

0 &

108

met

ers

per

seco

nd.H

ow lo

ng d

oes

it t

ake

light

from

the

sun

to

reac

h th

e ea

rth

in m

inut

es?

Rou

nd t

o th

e ne

ares

t hu

ndre

dth.

8.31

min

13.A

ligh

t-ye

ar is

the

dis

tanc

e th

at li

ght

trav

els

in o

ne y

ear.

(See

Exe

rcis

e 12

.) H

ow f

ar is

a li

ght

year

in k

ilom

eter

s? E

xpre

ss y

our

answ

er in

sci

enti

fic

nota

tion

.Rou

nd t

o th

e ne

ares

t hu

ndre

dth.

9.46

'10

12km

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



Ans

wer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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

Pol

ynom

ial

A p

olyn

omia

lis

a m

onom

ial o

r a

sum

of

mon

omia

ls.A

bin

omia

lis

the

sum

of

two

mon

omia

ls,a

nd a

tri

nom

ial

is t

he s

um o

f th

ree

mon

omia

ls.

Poly

nom

ials

wit

h m

ore

than

thr

ee t

erm

s ha

ve n

o sp

ecia

l nam

e.T

he d

egre

eof

a m

onom

ial

is t

he s

um o

f th

e ex

pone

nts

of a

ll it

s va

riab

les.

The

deg

ree

of t

he

pol

ynom

ial

is t

he s

ame

as t

he d

egre

e of

the

mon

omia

l ter

m w

ith

the

high

est

degr

ee.

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

Pol

ynom

ial?

Mon

omia

l,B

inom

ial,

Deg

ree

of t

he

or T

rino

mia

l?P

olyn

omia

l

3x$

7xyz

Yes.

3x

$7x

yz"

3x#

($7x

yz),

bino

mia

l3

whi

ch is

the

sum

of t

wo

mon

omia

ls

$25

Yes.

$25

is a

rea

l num

ber.

mon

omia

l0

7n3

#3n

$4

No.

3n$

4"

, whi

ch is

not

no

ne o

f the

se—

a m

onom

ial

Yes.

The

exp

ress

ion

sim

plifi

es to

9 x3

#4x

#x

#4

#2x

9x3

#7x

#4,

whi

ch is

the

sum

tr

inom

ial

3of

thre

e m

onom

ials

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;

mon

omia

l2.

#5

no

3.7x

$x

#5

yes;

bino

mia

l4.

8g2 h

$7g

h#

2ye

s;tr

inom

ial

5.#

5y$

8no

6.6x

#x2

yes;

bino

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

olyn

omia

ls in

Ord

erT

he t

erm

s of

a p

olyn

omia

l are

usu

ally

arr

ange

d so

tha

t th

e po

wer

s of

one

var

iabl

e ar

e in

asc

end

ing

(inc

reas

ing)

ord

er o

r d

esce

nd

ing

(dec

reas

ing)

ord

er.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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.8x

3 y!

y2)

6x2 y

)xy

2

$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!

x24x

5#

x4$

x2

b.!

6xy

)y3

!x2

y2)

x4y2

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

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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#

n22.

4by

#2b

$by

3.$

32

yes;

bino

mia

lye

s;bi

nom

ial

yes;

mon

omia

l

4.5.

5x2

$3x

$4

6.2c

2#

8c#

9 $

3

yes;

mon

omia

lno

yes;

trin

omia

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;

bino

mia

lye

s;tr

inom

ial

yes;

mon

omia

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

#n2

m4

#n2

6

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)

10bx

)4b

2 x3

)x

4

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!

52x

5!

6x3

)4x

)2

18.g

2 x$

3gx3

#7g

3#

4x2

19.$

11x2

y3#

6y$

2xy

#2x

4

!3g

x3

)4x

2)

g2 x

)7g

32x

4!

11x

2 y3

!2x

y)

6y

20.7

a2x2

#17

$a3

x3#

2ax

21.1

2rx3

#9r

6#

r2x

#8x

6

!a

3 x3

)7a

2 x2

)2a

x)

178x

6)

12rx

3)

r2x

)9r

6

22.M

ON

EYW

rite

a p

olyn

omia

l to

repr

esen

t th

e va

lue

of t

ten-

dolla

r bi

lls,f

fift

y-do

llar

bills

,and

hon

e-hu

ndre

d-do

llar

bills

.10

t)50

f)10

0h

23.G

RA

VIT

YT

he h

eigh

t ab

ove

the

grou

nd o

f a

ball

thro

wn

up w

ith

a ve

loci

ty o

f 96

fee

t pe

rse

cond

fro

m a

hei

ght

of 6

fee

t is

6 #

96t

$16

t2fe

et,w

here

tis

the

tim

e in

sec

onds

.A

ccor

ding

to

this

mod

el,h

ow h

igh

is t

he b

all a

fter

7 s

econ

ds?

Exp

lain

.!

106

ft;T

he h

eigh

t is

neg

ativ

e be

caus

e th

e m

odel

doe

s no

t ac

coun

t fo

r th

e ba

ll hi

ttin

g th

e gr

ound

whe

n th

e he

ight

is 0

fee

t.1 # 4

dab

b

1 % 5

Pra

ctic

e (A

vera

ge)

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-4

8-4



Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csPo

lyno

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill47

7G

lenc

oe A

lgeb

ra 1

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

ucti

on t

o L

esso

n 8-

4 at

the

top

of

page

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

coul

d yo

u ca

ll a

poly

nom

ial w

ith

just

one

ter

m?

a m

onom

ial

Read

ing

the

Less

on

1.W

hat

is t

he m

eani

ng o

f th

e pr

efix

es m

ono-

,bi-

,and

tri

-?

Mon

o- m

eans

one

,bi-

mea

ns t

wo,

and

tri-

mea

ns t

hree

.

2.W

rite

exa

mpl

es o

f w

ords

tha

t be

gin

wit

h th

e pr

efix

es m

ono-

,bi-

,and

tri

-.

Sam

ple

answ

er:m

onoc

ycle

(on

e w

heel

),bi

cycl

e (t

wo

whe

els)

,tri

cycl

e(t

hree

whe

els)

3.C

ompl

ete

the

tabl

e. mon

omia

lbi

nom

ial

trin

omia

lpo

lyno

mia

l with

mor

e th

an t

hree

ter

ms

Exa

mpl

e3r

2 t2x

2#

3x5x

2#

3x#

27s

2#

s4#

2s3

$s

#5

Num

ber

of T

erm

s1

23

5

4.W

hat

is t

he d

egre

e of

the

mon

omia

l 3xy

2 z?

4

5.W

hat

is t

he d

egre

e of

the

pol

ynom

ial 4

x4#

2x3 y

3#

y2#

14?

Exp

lain

how

you

fou

ndyo

ur a

nsw

er.

6;S

ince

0 )

4 %

4 ,4

x4

has

degr

ee 4

;sin

ce 0

)3

)3

%6,

2x3 y

3ha

sde

gree

6;y

2ha

s de

gree

2;a

nd 1

4 ha

s de

gree

0.T

he h

ighe

st d

egre

e of

thes

e te

rms

is 6

.

Hel

ping

You

Rem

embe

r

6.U

se a

dic

tion

ary

to f

ind

the

mea

ning

of

the

term

s as

cend

ing

and

desc

endi

ng.W

rite

the

irm

eani

ngs

and

then

des

crib

e a

situ

atio

n in

you

r ev

eryd

ay li

fe t

hat

rela

tes

to t

hem

.

asce

ndin

g:go

ing,

grow

ing,

or m

ovin

g up

war

d;de

scen

ding

:mov

ing

from

a hi

gher

to

a lo

wer

pla

ce;S

ampl

e an

swer

:clim

bing

sta

irs,

hiki

ng

©G

lenc

oe/M

cGra

w-H

ill47

8G

lenc

oe A

lgeb

ra 1

Poly

nom

ial F

unct

ions

Supp

ose

a lin

ear

equa

tion

suc

h as

23x

#y

"4

is

sol

ved

for

y.T

hen

an e

quiv

alen

t eq

uati

on,

y"

3x#

4,is

fou

nd.E

xpre

ssed

in t

his

way

,yis

a

func

tion

of x

,or

f(x)

"3x

#4.

Not

ice

that

the

ri

ght

side

of

the

equa

tion

is a

bin

omia

l of

degr

ee 1

.

Hig

her-

degr

ee p

olyn

omia

ls in

xm

ay a

lso

form

fu

ncti

ons.

An

exam

ple

is f

(x) "

x3#

1,w

hich

is

a p

olyn

omia

l fun

ctio

n of

deg

ree

3.Yo

u ca

n gr

aph

this

fun

ctio

n us

ing

a ta

ble

of o

rder

ed

pair

s,as

sho

wn

at t

he r

ight

.

For

eac

h o

f th

e fo

llow

ing

pol

ynom

ial

fun

ctio

ns,

mak

e a

tabl

eof

val

ues

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

$1 %

1 2%$

2 %3 8%

$1

0

01

12

1 %1 2%

4%3 8%

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-4

8-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Add

ing

and

Sub

trac

ting

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-5

8-5

©G

lenc

oe/M

cGra

w-H

ill47

9G

lenc

oe A

lgeb

ra 1

Lesson 8-5

Add

Pol

ynom

ials

To a

dd p

olyn

omia

ls,y

ou c

an g

roup

like

ter

ms

hori

zont

ally

or

wri

teth

em in

col

umn

form

,alig

ning

like

ter

ms

vert

ical

ly.L

ike

term

sar

e m

onom

ial t

erm

s th

atar

e ei

ther

iden

tica

l or

diff

er o

nly

in t

heir

coe

ffic

ient

s,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 li

ke t

erm

s.

(2x2

#x

$8)

#(3

x$

4x2

#2)

"[(

2x2

#($

4x2 )

] #

(x#

3x) #

[($

8) #

2)]

"$

2x2

#4x

$6.

The

sum

is $

2x2

#4x

$6.

Fin

d (

3x2

)5x

y) )

(xy

)2x

2 ).

Ver

tica

l M

eth

odA

lign

like

term

s in

col

umns

and

add

.

3x2

#5x

y(#

)2x

2#

xyP

ut th

e te

rms

in d

esce

ndin

g or

der.

5x2

#6x

y

The

sum

is 5

x2#

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)

2.(6

x#

9) #

(4x2

$7)

7a)

14x

2)

6x)

2

3.(6

xy#

2y#

6x) #

(4xy

$x)

4.(x

2#

y2) #

($x2

#y2

)

10xy

)5x

)2y

2y2

5.(3

p2$

2p#

3) #

(p2

$7p

#7)

6.(2

x2#

5xy

#4y

2 ) #

($xy

$6x

2#

2y2 )

4p2

!9p

)10

!4x

2)

4xy

)6y

2

7.(5

p#

2q) #

(2p2

$8q

#1)

8.(4

x2$

x#

4) #

(5x

#2x

2#

2)

2p2

)5p

!6q

)1

6x2

)4x

)6

9.(6

x2#

3x) #

(x2

$4x

$3)

10.(

x2#

2xy

#y2

) #(x

2$

xy$

2y2 )

7x2

!x

!3

2x2

)xy

!y2

11.(

2a$

4b$

c) #

($2a

$b

$4c

)12

.(6x

y2#

4xy)

#(2

xy$

10xy

2#

y2)

!5b

!5c

!4x

y2)

6xy

)y2

13.(

2p$

5q) #

(3p

#6q

) #(p

$q)

14.(

2x2

$6)

#(5

x2#

2) #

($x2

$7)

6p6x

2!

11

15.(

3z2

#5z

) #(z

2#

2z) #

(z$

4)16

.(8x

2#

4x#

3y2

#y)

#(6

x2$

x#

4y)

4z2 )

8z!

414

x2)

3x)

3y2 )

5y

©G

lenc

oe/M

cGra

w-H

ill48

0G

lenc

oe A

lgeb

ra 1

Subt

ract

Pol

ynom

ials

You

can

subt

ract

a p

olyn

omia

l by

addi

ng it

s ad

diti

ve in

vers

e.To

fin

d th

e ad

diti

ve in

vers

e of

a p

olyn

omia

l,re

plac

e ea

ch t

erm

wit

h it

s ad

diti

ve in

vers

e or

opp

osit

e.

Fin

d (

3x2

)2x

!6)

!(2

x)

x2)

3).

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Add

ing

and

Sub

trac

ting

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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.

The

n gr

oup

like

term

s.

(3x2

#2x

$6)

$(2

x#

x2#

3)"

(3x2

#2x

$6)

#[(

$2x

)#($

x2) #

($3)

]"

[3x2

#($

x2)]

#[2

x#

($2x

)] #

[$6

#($

3)]

"2x

2#

($9)

"2x

2$

9

The

dif

fere

nce

is 2

x2$

9.

Ver

tica

l M

eth

od

Alig

n lik

e te

rms

in c

olum

ns a

ndsu

btra

ct b

y ad

ding

the

add

itiv

e in

vers

e.

3x2

#2x

$6

($)

x2#

2x#

3

3x2

#2x

$6

(#)$

x2$

2x$

3

2x2

$9

The

dif

fere

nce

is 2

x2$

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) $

(6xy

$2x

)4.

(x2

#y2

) $($

x2#

y2)

3xy

) y

2x2

5.(6

p2#

4p#

5) $

(2p2

$5p

#1)

6.(6

x2#

5xy

$2y

2 ) $

($xy

$2x

2$

4y2 )

4p2

)9p

)4

8x2

)6x

y)

2y2

7.(8

p$

5q) $

($6p

2#

6q$

3)8.

(8x2

$4x

$3)

$($

2x$

x2#

5)

6p2

)8p

!11

q)

39x

2!

2x!

8

9.(3

x2$

2x) $

(3x2

#5x

$1)

10.(

4x2

#6x

y#

2y2 )

$($

x2#

2xy

$5y

2 )

!7x

)1

5x2 )

4xy

)7y

2

11.(

2h$

6j$

2k) $

($7h

$5j

$4k

)12

.(9x

y2#

5xy)

$($

2xy

$8x

y2)

9h!

j)2k

17xy

2)

7xy

13.(

2a$

8b) $

($3a

#5b

)14

.(2x

2$

8) $

($2x

2$

6)

5a!

13b

4x2

!2

15.(

6z2

#4z

#2)

$(4

z2#

z)16

.(6x

2$

5x#

1) $

($7x

2$

2x#

4)

2z2

)3z

)2

13x2

!3x

!3



Ans

wer

s

Skil

ls P

ract

ice

Add

ing

and

Sub

trac

ting

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill48

1G

lenc

oe A

lgeb

ra 1

Lesson 8-5

Fin

d e

ach

su

m o

r d

iffe

ren

ce.

1.(2

x#

3y) #

(4x

#9y

) 6x

)12

y2.

(6s

#5t

) #(4

t#

8s) 1

4s)

9t

3.(5

a#

9b) $

(2a

#4b

) 3a

)5b

4.(1

1m$

7n) $

(2m

#6n

) 9m

!13

n

5.(m

2$

m) #

(2m

#m

2 ) 2

m2

)m

6.(x

2$

3x) $

(2x2

#5x

) !x2

!8x

7.(d

2$

d#

5) $

(2d

#5)

d2

!3d

8.(2

e2$

5e) #

(7e

$3e

2 ) !

e2)

2e

9.(5

f#

g$

2) #

($2f

#3)

10.(

6k2

#2k

#9)

#(4

k2$

5k)

3f)

g)

110

k2

!3k

)9

11.(

x3$

x#

1) $

(3x

$1)

12.(

b2#

ab$

2) $

(2b2

#2a

b)

x3

!4x

)2

!b

2!

ab!

2

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7z2

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Fin

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27.B

USI

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he p

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00 m

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ofit

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EOM

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The

mea

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tw

o si

des

of a

tri

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e gi

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is t

he p

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eter

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vera

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ing

and

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trac

ting

Poly

nom

ials

NA

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____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-5

8-5



Rea

din

g t

o L

earn

Math

emati

csA

ddin

g an

d S

ubtr

actin

g Po

lyno

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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RIO

D__

___

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

ucti

on t

o L

esso

n 8-

5 at

the

top

of

page

439

in y

our

text

book

.

Wha

t op

erat

ion

wou

ld y

ou u

se t

o fi

nd h

ow m

uch

mor

e th

e tr

adit

iona

l toy

sale

s R

wer

e th

an t

he v

ideo

gam

es s

ales

V?

subt

ract

ion

Read

ing

the

Less

on

1.U

se t

he e

xam

ple

($3x

3#

4x2

#5x

#1)

#($

5x3

$2x

2#

2x$

7).

a.Sh

ow w

hat

is m

eant

by

grou

ping

like

ter

ms

hori

zont

ally

.

[!3x

3)

(!5x

3 )]

)[(

4x2

)(!

2x2 )

] )

(5x

)2x

) )

[1 )

(!7)

]

b.Sh

ow w

hat

is m

eant

by

alig

ning

like

ter

ms

vert

ical

ly.

!3x

3)

4x2

)5x

)1

())

!5x

3!

2x2

)2x

!7

c.C

hoos

e on

e m

etho

d,th

en a

dd t

he p

olyn

omia

ls.

!8x

3)

2x2

)7x

!6

2.H

ow is

sub

trac

ting

a p

olyn

omia

l lik

e su

btra

ctin

g a

rati

onal

num

ber?

You

subt

ract

by

addi

ng t

he a

dditi

ve in

vers

e.

3.A

n al

gebr

a st

uden

t go

t th

e fo

llow

ing

exer

cise

wro

ng o

n hi

s ho

mew

ork.

Wha

t w

as

his

erro

r?

(3x5

$3x

4#

2x3

$4x

2#

5) $

(2x5

$x3

#2x

2$

4)"

[3x5

#($

2x5 )

] #($

3x4 )

#[2

x3#

($x3

)] #

[$4x

2#

($2x

2 )] #

(5 #

4)"

x5$

3x4

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2#

9

He

did

not

add

the

addi

tive

inve

rse

of !

x3 .

Hel

ping

You

Rem

embe

r

4.H

ow is

add

ing

and

subt

ract

ing

poly

nom

ials

ver

tica

lly li

ke a

ddin

g an

d su

btra

ctin

gde

cim

als

vert

ical

ly?

Alig

ning

like

ter

ms

whe

n ad

ding

or

subt

ract

ing

poly

nom

ials

is li

ke u

sing

plac

e va

lue

to a

lign

digi

ts w

hen

you

add

or s

ubtr

act

deci

mal

s.

©G

lenc

oe/M

cGra

w-H

ill48

4G

lenc

oe A

lgeb

ra 1

Cir

cula

r A

reas

and

Vol

umes

Are

a of

Cir

cle

Vol

um

e of

Cyl

inde

rV

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

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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

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!in

33!

x3

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!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

xx

r

h

r

h

r

1 % 3

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-5

8-5



Ans

wer

s

Stu

dy

Gu

ide

an

d I

nte

rven

tion

Mul

tiply

ing

a Po

lyno

mia

l by

a M

onom

ial

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill48

5G

lenc

oe A

lgeb

ra 1

Lesson 8-6

Prod

uct

of M

onom

ial a

nd P

olyn

omia

lT

he D

istr

ibut

ive

Pro

pert

y ca

n be

use

d to

mul

tipl

y a

poly

nom

ial b

y a

mon

omia

l.Yo

u ca

n m

ulti

ply

hori

zont

ally

or

vert

ical

ly.S

omet

imes

mul

tipl

ying

res

ults

in li

ke t

erm

s.T

he p

rodu

cts

can

be s

impl

ifie

d by

com

bini

ng 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) !

(!24

x2)

#!

12x4

!18

x3"

24x2

Ver

tica

l M

eth

od4x

2"

6x!

8($

) !

3x2

!12

x4!

18x3

"24

x2

The

pro

duct

is !

12x4

!18

x3"

24x2

.

Sim

pli

fy !

2(4x

2"

5x)

!x(

x2"

6x).

!2(

4x2

"5x

) !x(

x2

"6x

)#

!2(

4x2 )

"(!

2)(5

x) "

(!x)

(x2 )

"(!

x)(6

x)#

!8x

2"

(!10

x) "

(!x3

) "(!

6x2 )

#(!

x3) "

[!8x

2"

(!6x

2 )] "

(!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) !

6x2

3x2

!9x

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3!

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a(2a

!b)

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2r2

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4a2

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b32

r3"

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n(3n

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n(3

!n)

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b(b2

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)

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n!

7b3

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b2

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b

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2z(4

z2!

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1) !

z(3z

2"

2z!

1)17

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x2!

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3(!

6x2

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(x!

1)

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z3

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x2!

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©G

lenc

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w-H

ill48

6G

lenc

oe A

lgeb

ra 1

Solv

e Eq

uati

ons

wit

h Po

lyno

mia

l Exp

ress

ions

Man

y eq

uati

ons

cont

ain

poly

nom

ials

tha

t m

ust

be a

dded

,sub

trac

ted,

or m

ulti

plie

d be

fore

the

equ

atio

n ca

n be

sol

ved.

Sol

ve 4

(n!

2) "

5n#

6(3

!n

) "

19.

4(n

!2)

"5n

#6(

3 !

n) "

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rigin

al e

quat

ion

4n!

8 "

5n#

18 !

6n"

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istr

ibut

ive

Pro

pert

y

9n!

8 #

37 !

6nC

ombi

ne li

ke te

rms.

15n

!8

#37

Add

6n

to b

oth

side

s.

15n

#45

Add

8 to

bot

h si

des.

n#

3D

ivid

e ea

ch s

ide

by 1

5.

The

sol

utio

n is

3.

Sol

ve e

ach

equ

atio

n.

1.2(

a!

3) #

3(!

2a"

6)3

2.3(

x"

5) !

6 #

183

3.3x

(x!

5) !

3x2

#!

302

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x2"

2x) #

2(3x

2"

12)

2

5.4(

2p"

1) !

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8p"

12)

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6.2(

6x"

4) "

2 #

4(x

!4)

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2(4y

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"6

#4(

y!

2)1

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c"

2) !

c(c

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c!

126

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x2!

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3x2

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110

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x"

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2 #

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x"

1)!

1

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97

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0x2

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(x"

2)!

3

15.3

(x"

2) "

2(x

"1)

#!

5(x

!3)

16.4

(3p2

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) !12

p2#

2(8p

"6)

!3 $ 2

7 $ 10

1 $

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Mul

tiply

ing

a P

olyn

omia

l by

a M

onom

ial

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises

3 ! 5



Skil

ls P

ract

ice

Mul

tiply

ing

a Po

lyno

mia

l by

a M

onom

ial

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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)

$5p

16.y

2 ($

4y#

5) $

6y2

!2p

2)

3p!

4y3

!y

2

17.2

x(3x

2#

4) $

3x3

18.4

a(5a

2$

4) #

9a

3x3

)8x

20a3

!7a

19.4

b($

5b$

3) $

2(b2

$7b

$4)

20.3

m(3

m#

6) $

3(m

2#

4m#

1)

!22

b2

)2b

)8

6m2

)6m

!3

Sol

ve e

ach

equ

atio

n.

21.3

(a#

2) #

5 "

2a#

4!

722

.2(4

x#

2) $

8 "

4(x

#3)

4

23.5

(y#

1) #

2 "

4(y

#2)

$6

!5

24.4

(b#

6) "

2(b

#5)

#2

!6

25.6

(m$

2) #

14 "

3(m

#2)

$10

!2

26.3

(c#

5) $

2 "

2(c

#6)

#2

1

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lenc

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8G

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lgeb

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Fin

d e

ach

pro

du

ct.

1.2h

($7h

2$

4h)

2.6p

q(3p

2#

4q)

3.$

2u2 n

(4u

$2n

)!

14h

3!

8h2

18p

3 q)

24pq

2!

8u3 n

)4u

2 n2

4.5j

k(3j

k#

2k)

5.$

3rs(

$2s

2#

3r)

6.4m

g2(2

mg

#4g

)15

j2k

2)

10jk

26r

s3!

9r2 s

8m2 g

3)

16m

g3

7.$

m(8

m2

#m

$7)

8.$

n2($

9n2

#3n

#6)

!2m

3!

m2

)m

6n4

!2n

3!

4n2

Sim

pli

fy.

9.$

2!(3

!$

4) #

7!10

.5w

($7w

#3)

#2w

($2w

2#

19w

#2)

!6!

2)

15!

!4w

3)

3w2

)19

w

11.6

t(2t

$3)

$5(

2t2

#9t

$3)

12.$

2(3m

3#

5m#

6) #

3m(2

m2

#3m

#1)

2t2

!63

t)15

9m2

!7m

!12

13.$

3g(7

g$

2) #

3(g2

#2g

#1)

$3g

($5g

#3)

!3g

2)

3g)

3

14.4

z2(z

$7)

$5z

(z2

$2z

$2)

#3z

(4z

$2)

!z3

!6z

2)

4z

Sol

ve e

ach

equ

atio

n.

15.5

(2s

$1)

#3

"3(

3s#

2)8

16.3

(3u

#2)

#5

"2(

2u$

2)!

3

17.4

(8n

#3)

$5

"2(

6n#

8) #

118

.8(3

b#

1) "

4(b

#3)

$9

!

19.h

(h$

3) $

2h"

h(h

$2)

$12

420

.w(w

#6)

#4w

"$

7 #

w(w

#9)

!7

21.t

(t#

4) $

1 "

t(t

#2)

#2

22.u

(u$

5) #

8u"

u(u

#2)

$4

!4

23.N

UM

BER

TH

EORY

Let

xbe

an

inte

ger.

Wha

t is

the

pro

duct

of

twic

e th

e in

tege

r ad

ded

to t

hree

tim

es t

he n

ext

cons

ecut

ive

inte

ger?

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

.He

allo

cate

d x

dolla

rs o

f th

e m

oney

to

a bo

ndac

coun

t th

at e

arns

4%

inte

rest

per

yea

r an

d th

e re

st t

o a

trad

itio

nal a

ccou

nt t

hat

earn

s 5%

inte

rest

per

yea

r.

24.W

rite

an

expr

essi

on t

hat

repr

esen

ts t

he a

mou

nt o

f m

oney

inve

sted

in t

he t

radi

tion

alac

coun

t.5,

000

!x

25.W

rite

a p

olyn

omia

l mod

el in

sim

ples

t fo

rm f

or t

he t

otal

am

ount

of

mon

ey T

Ken

t ha

sin

vest

ed a

fter

one

yea

r.(H

int:

Eac

h ac

coun

t ha

s A

#IA

dolla

rs,w

here

Ais

the

ori

gina

lam

ount

in t

he a

ccou

nt a

nd I

is it

s in

tere

st r

ate.

)T

%5,

250

!0.

01x

26.I

f K

ent

put

$500

in t

he b

ond

acco

unt,

how

muc

h m

oney

doe

s he

hav

e in

his

ret

irem

ent

plan

aft

er o

ne y

ear?

$5,2

45

3 # 2

1 # 41 # 2

7 # 41 # 4

2 % 31 % 4

Pra

ctic

e (A

vera

ge)

Mul

tiply

ing

a Po

lyno

mia

l by

a M

onom

ial

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6



Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csM

ultip

lyin

g a

Poly

nom

ial b

y a

Mon

omia

l

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill48

9G

lenc

oe A

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

ucti

on t

o L

esso

n 8-

6 at

the

top

of

page

444

in y

our

text

book

.

You

may

rec

all t

hat

the

form

ula

for

the

area

of

a re

ctan

gle

is A

"!w

.In

this

rec

tang

le,!

"an

d w

".H

ow w

ould

you

subs

titu

te t

hese

val

ues

in t

he a

rea

form

ula?

A%

(x)

3)(2

x)

Read

ing

the

Less

on

1.R

efer

to

Les

son

8-6.

a.H

ow is

the

Dis

trib

utiv

e P

rope

rty

used

to

mul

tipl

y a

poly

nom

ial b

y a

mon

omia

l?

The

mon

omia

l is

mul

tiplie

d by

eac

h te

rm in

the

pol

ynom

ial.

b.U

se t

he D

istr

ibut

ive

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 t

he d

iffe

renc

e be

twee

n si

mpl

ifyi

ng a

n ex

pres

sion

and

sol

ving

an

equa

tion

?

Sim

plify

ing

an e

xpre

ssio

n is

com

bini

ng li

ke t

erm

s.S

olvi

ng a

n eq

uatio

nis

fin

ding

the

val

ue o

f th

e va

riab

le t

hat

mak

es t

he e

quat

ion

true

.

Hel

ping

You

Rem

embe

r

3.U

se t

he e

quat

ion

2x(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

ving

equ

atio

ns w

ith

poly

nom

ial e

xpre

ssio

ns t

o an

othe

r al

gebr

a st

uden

t.

Use

the

Dis

trib

utiv

e P

rope

rty.

2x2

!10

x)

3x2

)9x

%5x

2)

35x

!9

Com

bine

like

ter

ms.

5 x2

!x

%5x

2)

35x

!9

Sub

trac

t 5 x

2fr

om b

oth

side

s.!

x%

35x

!9

Sub

trac

t 35

xfr

om b

oth

side

s.!

36x

%!

9

Div

ide

each

sid

e by

!36

.x

%0.

25

9x3

6x5

!3x

6

(!3x

3 )(3

)(!

3x3 )

(2x

2 )!

3x3 (

x3 )

14y

24y

36y

4

72y

3y2

2xx

)3

©G

lenc

oe/M

cGra

w-H

ill49

0G

lenc

oe A

lgeb

ra 1

Figu

rate

Num

bers

The

num

bers

bel

ow a

re c

alle

d pe

nta

gon

al n

um

bers

.The

y ar

e th

e nu

mbe

rs o

f dot

s or

dis

ksth

at c

an b

e ar

rang

ed a

s pe

ntag

ons.

1.F

ind

the

prod

uct

n(3n

$1)

.!

2.E

valu

ate

the

prod

uct

in E

xerc

ise

1 fo

r va

lues

of n

from

1 t

hrou

gh 4

.1,

5,12

,22

3.W

hat

do y

ou n

otic

e?Th

ey a

re t

he f

irst

four

pen

tago

nal n

umbe

rs.

4.F

ind

the

next

six

pen

tago

nal n

umbe

rs.

35,5

1,70

,92,

117,

145

5.F

ind

the

prod

uct

n(n

#1)

.)

6.E

valu

ate

the

prod

uct

in E

xerc

ise

5 fo

r va

lues

of n

from

1 t

hrou

gh 5

.On

anot

her

shee

t of

pape

r,m

ake

draw

ings

to

show

why

the

se n

umbe

rs a

re c

alle

d th

e tr

iang

ular

num

bers

.1,

3,6,

10,1

5

7.F

ind

the

prod

uct

n(2n

$1)

.2n

2!

n

8.E

valu

ate

the

prod

uct

in E

xerc

ise

7 fo

r va

lues

of n

from

1 t

hrou

gh 5

.Dra

w t

hese

hexa

gona

l num

bers

.1,

6,15

,28,

45

9.F

ind

the

firs

t 5

squa

re n

umbe

rs.A

lso,

wri

te t

he g

ener

al e

xpre

ssio

n fo

r an

y sq

uare

num

ber.

1,4,

9,16

,25;

n2

The

num

bers

you

hav

e ex

plor

ed a

bove

are

cal

led

the

plan

e fi

gura

te n

umbe

rs b

ecau

se t

hey

can

be a

rran

ged

to m

ake

geom

etri

c fi

gure

s.Yo

u ca

n al

so c

reat

e so

lid f

igur

ate

num

bers

.

10.I

f yo

u pi

le 1

0 or

ange

s in

to a

pyr

amid

wit

h a

tria

ngle

as

a ba

se,y

ou g

et o

ne o

f th

ete

trah

edra

l num

bers

.How

man

y la

yers

are

the

re in

the

pyr

amid

? H

ow m

any

oran

ges

are

ther

e in

the

bot

tom

laye

rs?

3 la

yers

;6

11.E

valu

ate

the

expr

essi

on

n3#

n2#

nfo

r va

lues

of n

from

1 t

hrou

gh 5

to

find

the

firs

t fi

ve t

etra

hedr

al n

umbe

rs.

1,4,

10,2

0,35

1 % 31 % 2

1 % 6

n # 2n2# 2

1 % 2

n # 23 n

2#

21 % 2

●●

2212

51

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6



Stu

dy

Gu

ide

and I

nte

rven

tion

Mul

tiply

ing

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill49

1G

lenc

oe A

lgeb

ra 1

Lesson 8-7

Mul

tipl

y Bi

nom

ials

To m

ulti

ply

two

bino

mia

ls,y

ou c

an a

pply

the

Dis

trib

utiv

e P

rope

rty

twic

e.A

use

ful w

ay t

o ke

ep t

rack

of

term

s in

the

pro

duct

is t

o us

e th

e F

OIL

met

hod

asill

ustr

ated

in E

xam

ple

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)

#3(

x)#

3($

4)"

x2$

4x#

3x$

12"

x2$

x$

12

Ver

tica

l M

eth

od

x#

3(&

) x

$4

$4x

$12

x2#

3xx2

$x

$12

The

pro

duct

is x

2$

x$

12.

Fin

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)

#($

2)(x

) #($

2)(5

)"

x2#

5x#

($2x

) $10

"x2

#3x

$10

The

pro

duct

is x

2#

3x$

10.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d e

ach

pro

du

ct.

1.(x

#2)

(x#

3)2.

(x$

4)(x

#1)

3.(x

$6)

(x$

2)

x2)

5x)

6x2

!3x

!4

x2!

8x)

12

4.(p

$4)

(p#

2)5.

(y#

5)(y

#2)

6.(2

x$

1)(x

#5)

p2

!2p

!8

y2)

7y)

102x

2)

9x!

5

7.(3

n$

4)(3

n$

4)8.

(8m

$2)

(8m

#2)

9.(k

#4)

(5k

$1)

9n2

!24

n)

1664

m2

!4

5k2

)19

k!

4

10.(

3x#

1)(4

x#

3)11

.(x

$8)

($3x

#1)

12.(

5t#

4)(2

t$

6)

12x2

)13

x)

3!

3x2

)25

x!

810

t2!

22t!

24

13.(

5m$

3n)(

4m$

2n)

14.(

a$

3b)(

2a$

5b)

15.(

8x$

5)(8

x#

5)

20m

2!

22m

n)

6n2

2a2

!11

ab)

15b

264

x2!

25

16.(

2n$

4)(2

n#

5)17

.(4m

$3)

(5m

$5)

18.(

7g$

4)(7

g#

4)

4n2

)2n

!20

20m

2!

35m

)15

49g

2!

16

©G

lenc

oe/M

cGra

w-H

ill49

2G

lenc

oe A

lgeb

ra 1

Mul

tipl

y Po

lyno

mia

lsT

he D

istr

ibut

ive

Pro

pert

y ca

n be

use

d to

mul

tipl

y an

y tw

o po

lyno

mia

ls. F

ind

(3x

)2)

(2x2

!4x

)5)

.

(3x

#2)

(2x2

$4x

#5)

"3x

(2x2

$4x

#5)

#2(

2x2

$4x

#5)

Dis

trib

utiv

e P

rope

rty

"6x

3$

12x2

#15

x#

4x2

$8x

#10

Dis

trib

utiv

e P

rope

rty

"6x

3$

8x2

#7x

#10

Com

bine

like

term

s.

The

pro

duct

is 6

x3$

8x2

#7x

#10

.

Fin

d e

ach

pro

du

ct.

1.(x

#2)

(x2

$2x

#1)

2.(x

#3)

(2x2

#x

$3)

x3

!3x

)2

2x3

)7x

2!

9

3.(2

x$

1)(x

2$

x#

2)4.

(p$

3)(p

2$

4p#

2)

2x3

!3x

2)

5x!

2p

3!

7p2

)14

p!

6

5.(3

k#

2)(k

2#

k$

4)6.

(2t

#1)

(10t

2$

2t$

4)

3k3

)5k

2!

10k

!8

20t3

)6t

2!

10t!

4

7.(3

n$

4)(n

2#

5n$

4)8.

(8x

$2)

(3x2

#2x

$1)

3n3

)11

n2

!32

n)

1624

x3

)10

x2!

12x

)2

9.(2

a#

4)(2

a2$

8a#

3)10

.(3x

$4)

(2x2

#3x

#3)

4a3

!8a

2!

26a

)12

6x3

)x2

!3x

!12

11.(

n2#

2n$

1)(n

2#

n#

2)12

.(t2

#4t

$1)

(2t2

$t

$3)

n4

)3n

3)

3n2

)3n

!2

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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Mul

tiply

ing

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Ans

wer

s

Skil

ls P

ract

ice

Mul

tiply

ing

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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 un

its2

(x2

)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)

9h2

16x2

)8x

y)

y2

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)

(2d

2#

5d$

2)4h

3)

12h

2)

17h

)12

6d3

)21

d2

)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

units

230

.4x

2)

3x!

1 un

its2

31.N

UM

BER

TH

EORY

Let

xbe

an

even

inte

ger.

Wha

t is

the

pro

duct

of

the

next

tw

oco

nsec

utiv

e ev

en in

tege

rs?

x2)

6x)

8

32.G

EOM

ETRY

The

vol

ume

of a

rec

tang

ular

pyr

amid

is o

ne t

hird

the

pro

duct

of

the

area

of it

s ba

se a

nd it

s he

ight

.Fin

d an

exp

ress

ion

for

the

volu

me

of a

rec

tang

ular

pyr

amid

who

se b

ase

has

an a

rea

of 3

x2#

12x

#9

squa

re f

eet

and

who

se 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 (A

vera

ge)

Mul

tiply

ing

Poly

nom

ials

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7



Rea

din

g t

o L

earn

Math

emati

csM

ultip

lyin

g Po

lyno

mia

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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

bers

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

7 at

the

top

of

page

452

in y

our

text

book

.

In y

our

own

wor

ds,e

xpla

in h

ow t

he d

istr

ibut

ive

prop

erty

is u

sed

twic

e to

mul

tipl

y tw

o-di

git

num

bers

.

The

ones

of

the

first

fac

tor

are

mul

tiplie

d by

the

ten

s an

d th

eon

es o

f th

e ot

her

fact

or.T

hen

the

tens

of

the

first

fac

tor

are

mul

tiplie

d by

the

ten

s an

d on

es o

f th

e ot

her

fact

or.

Read

ing

the

Less

on

1.H

ow is

mul

tipl

ying

bin

omia

ls s

imila

r to

mul

tipl

ying

tw

o-di

git

num

bers

?

Bin

omia

ls h

ave

two

term

s an

d ea

ch t

erm

of

one

bino

mia

l is

mul

tiplie

d by

each

ter

m o

f th

e ot

her

bino

mia

l.

2.C

ompl

ete

the

tabl

e us

ing

the

FO

IL m

etho

d.

Pro

duct

of

)P

rodu

ct o

f)

Pro

duct

of

)P

rodu

ct o

f Fi

rst T

erm

sO

uter

Ter

ms

Inne

r Ter

ms

Last

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

ping

You

Rem

embe

r

3.T

hink

of

a m

etho

d fo

r re

mem

beri

ng a

ll th

e pr

oduc

t co

mbi

nati

ons

used

in t

he F

OIL

met

hod

for

mul

tipl

ying

tw

o bi

nom

ials

.Des

crib

e yo

ur m

etho

d us

ing

wor

ds o

r a

diag

ram

.S

ampl

e an

swer

:Im

agin

e th

at t

he t

wo

bino

mia

ls a

re w

ritt

en o

n th

e flo

or.

For

FOIL

,thi

nk o

f al

l the

pos

sibl

e w

ays

you

coul

d ha

ve y

our

left

foot

on

a te

rm o

f th

e fir

st b

inom

ial a

nd y

our

righ

t fo

ot o

n a

term

of

the

seco

ndbi

nom

ial.

Your

fee

t co

uld

be o

n th

e fir

st t

erm

s,th

e ou

ter

term

s,th

e in

ner

term

s,or

the

last

ter

ms.

©G

lenc

oe/M

cGra

w-H

ill49

6G

lenc

oe A

lgeb

ra 1

Pasc

al’s

Tri

angl

eT

his

arr

ange

men

t of

nu

mbe

rs i

s ca

lled

Pas

cal’s

Tri

angl

e.It

was

fir

st p

ubl

ish

ed i

n 1

665,

but

was

kn

own

hu

nd

red

s of

yea

rs e

arli

er.

1.E

ach

num

ber

in t

he t

rian

gle

is f

ound

by

addi

ng t

wo

num

bers

.W

hat

two

num

bers

wer

e ad

ded

to g

et t

he 6

in t

he 5

th r

ow?

3 an

d 3

2.D

escr

ibe

how

to

crea

te t

he 6

th r

ow o

f Pa

scal

’s T

rian

gle.

The

first

and

last

num

bers

are

1.E

valu

ate

1 )

4,4

)6,

6 )

4,an

d 4

)1

to f

ind

the

othe

r nu

mbe

rs.

3.W

rite

the

num

bers

for

row

s 6

thro

ugh

10 o

f th

e tr

iang

le.

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 th

e ex

pand

ed f

orm

of

(a#

b)n

and

Pasc

al’s

Tri

angl

e.

The

coef

ficie

nts

of t

he e

xpan

ded

form

are

foun

d in

row

n)

1 of

Pas

cal’s

Tria

ngle

.

8.U

se P

asca

l’s T

rian

gle

to w

rite

the

exp

ande

d fo

rm o

f (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

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7



Ans

wer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Spe

cial

Pro

duct

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-8

8-8

©G

lenc

oe/M

cGra

w-H

ill49

7G

lenc

oe A

lgeb

ra 1

Lesson 8-8

Squa

res

of S

ums

and

Dif

fere

nces

Som

e pa

irs

of b

inom

ials

hav

e pr

oduc

ts t

hat

follo

w s

peci

fic

patt

erns

.One

suc

h pa

tter

n is

cal

led

the

squa

re o

f a

sum

.Ano

ther

is c

alle

d th

esq

uare

of

a di

ffer

ence

.

Squ

are

of a

sum

(a#

b)2

"(a

#b)

(a#

b) "

a2#

2ab

#b2

Squ

are

of a

diff

eren

ce(a

$b)

2"

(a$

b)(a

$b)

"a2

$2a

b#

b2

Fin

d (

3a)

4)(3

a)

4).

Use

the

squ

are

of a

sum

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

The

pro

duct

is 9

a2#

24a

#16

.

Fin

d (

2z!

9)(2

z!

9).

Use

the

squ

are

of a

dif

fere

nce

patt

ern

wit

ha

"2z

and

b"

9.

(2z

$9)

(2z

$9)

"(2

z)2

$2(

2z)(

9) #

(9)(

9)"

4z2

$36

z#

81

The

pro

duct

is 4

z2$

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)

16pq

)16

q2

x2!

x)

48 # 3

4 # 92 % 3

3 # 21 # 161 % 4

©G

lenc

oe/M

cGra

w-H

ill49

8G

lenc

oe A

lgeb

ra 1

Prod

uct

of a

Sum

and

a D

iffe

renc

eT

here

is a

lso

a pa

tter

n fo

r th

e pr

oduc

t of

asu

m a

nd a

dif

fere

nce

of t

he s

ame

two

term

s,(a

#b)

(a$

b).T

he p

rodu

ct is

cal

led

the

dif

fere

nce

of

squ

ares

.

Pro

duct

of

a S

um a

nd a

Diff

eren

ce(a

#b)

(a$

b) "

a2$

b2

Fin

d (

5x)

3y)(

5x!

3y).

(a#

b)(a

$b)

"a2

$b2

Pro

duct

of a

Sum

and

a D

iffer

ence

(5x

#3y

)(5x

$3y

) "(5

x)2

$(3

y)2

a"

5xan

d b

"3y

"25

x2$

9y2

Sim

plify

.

The

pro

duct

is 2

5x2

$9y

2 .

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

.(h2

$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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Spe

cial

Pro

duct

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-8

8-8

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Spe

cial

Pro

duct

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-8

8-8

©G

lenc

oe/M

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

The

leng

th o

f a

rect

angl

e is

the

sum

of

two

who

le n

umbe

rs.T

he w

idth

of

the

rect

angl

e is

the

dif

fere

nce

of t

he s

ame

two

who

le n

umbe

rs.U

sing

the

se f

acts

,wri

te a

verb

al e

xpre

ssio

n fo

r th

e ar

ea o

f th

e re

ctan

gle.

The

area

is t

he s

quar

e of

the

larg

er n

umbe

r m

inus

the

squ

are

of t

he s

mal

ler

num

ber.

©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!

14up

)49

p2

22.(

4b$

7v)2

23.(

6n#

4p)2

24.(

5q#

6s)2

16b

2!

56bv

)49

v2

36n

2)

48np

)16

p2

25q

2)

60qs

)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

Jane

lle w

ants

to

enla

rge

a sq

uare

gra

ph t

hat

she

has

mad

e so

tha

t a

side

of t

he n

ew g

raph

will

be

1 in

ch m

ore

than

tw

ice

the

orig

inal

sid

e s.

Wha

t tr

inom

ial

repr

esen

ts t

he a

rea

of t

he e

nlar

ged

grap

h?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

uine

a pi

g,pu

re b

lack

hai

r co

lori

ng B

is d

omin

ant

over

pur

e w

hite

col

orin

g b.

Supp

ose

two

hybr

id B

bgu

inea

pig

s,w

ith

blac

k ha

ir c

olor

ing,

are

bred

.

35.F

ind

an e

xpre

ssio

n fo

r th

e ge

neti

c m

ake-

up o

f the

gui

nea

pig

offs

prin

g.0.

25B

B)

0.50

Bb

)0.

25bb

36.W

hat

is t

he p

roba

bilit

y th

at t

wo

hybr

id g

uine

a pi

gs w

ith

blac

k ha

ir c

olor

ing

will

pro

duce

a gu

inea

pig

wit

h w

hite

hai

r co

lori

ng?

25%

Pra

ctic

e (A

vera

ge)

Spe

cial

Pro

duct

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-8

8-8



Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csS

peci

al P

rodu

cts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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 bi

nom

ials

als

o a

bin

omia

l?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

8 at

the

top

of

page

458

in y

our

text

ook.

Wha

t is

a m

eant

by

the

term

tri

nom

ial

prod

uct?

a th

ree-

term

pol

ynom

ial a

nsw

er w

hen

mul

tiply

ing

poly

nom

ials

Read

ing

the

Less

on

1.R

efer

to

the

Key

Con

cept

s bo

xes

on p

ages

458

,459

,and

460

.

a.W

hen

mul

tipl

ying

tw

o bi

nom

ials

,the

re a

re t

hree

spe

cial

pro

duct

s.W

hat

are

the

thre

esp

ecia

l pro

duct

s th

at m

ay r

esul

t w

hen

mul

tipl

ying

tw

o bi

nom

ials

?

squa

re o

f a

sum

,squ

are

of a

diff

eren

ce,p

rodu

ct o

f a

sum

and

adi

ffer

ence

b.E

xpla

in w

hat

is m

eant

by

the

nam

e of

eac

h sp

ecia

l pro

duct

.

squa

re o

f a

sum

:squ

arin

g th

e su

m o

f tw

o m

onom

ials

;squ

are

of a

diff

eren

ce:s

quar

ing

the

diff

eren

ce o

f tw

o m

onom

ials

;pro

duct

of

asu

m a

nd a

diff

eren

ce:t

he p

rodu

ct o

f th

e su

m a

nd t

he d

iffer

ence

of

the

sam

e tw

o te

rms

c.U

se t

he e

xam

ples

in t

he K

ey C

once

pts

boxe

s to

com

plet

e th

e ta

ble.

Sym

bols

Pro

duct

Exa

mpl

eP

rodu

ct

Squ

are

of a

Sum

(a!

b)2

a2!

2ab

!b

2(x

!7)

2x2

!14

x!

49

Squ

are

of a

D

iffer

ence

(a"

b)2

a2"

2ab

!b

2(x

"4)

2x2

"8x

!16

Pro

duct

of

a S

um

and

a D

iffer

ence

(a!

b)(a

"b)

a2"

b2

(x!

9)(x

"9)

x2"

81

2.W

hat

is a

noth

er p

hras

e th

at d

escr

ibes

the

pro

duct

of

the

sum

and

dif

fere

nce

of

two

term

s?di

ffer

ence

of

squa

res

Hel

ping

You

Rem

embe

r

3.E

xpla

in h

ow F

OIL

can

hel

p yo

u re

mem

ber

how

man

y te

rms

are

in t

he s

peci

al p

rodu

cts

stud

ied

in t

his

less

on.

For

the

squa

re o

f a

sum

and

the

squ

are

of a

diff

eren

ce,t

he in

ner

and

oute

r pr

oduc

ts a

re e

qual

,so

ther

e ar

e ar

e on

lyth

ree

term

s.Fo

r th

e pr

oduc

t of

the

sum

and

diff

eren

ce o

f tw

o te

rms,

two

of t

he p

rodu

cts

for

FOIL

are

opp

osite

s.Th

at m

eans

tha

t th

e fin

al p

rodu

ctha

s on

ly t

wo

term

s.

©G

lenc

oe/M

cGra

w-H

ill50

2G

lenc

oe A

lgeb

ra 1

Sum

s an

d D

iffer

ence

s of

Cub

es

Rec

all t

he f

orm

ulas

for

fin

ding

som

e sp

ecia

l pro

duct

s:

Perf

ect-

squa

re t

rino

mia

ls:

(a!

b)2

"a2

!2a

b!

b2or

(a

#b)

2"

a2#

2ab

!b2

Dif

fere

nce

of t

wo

squa

res:

(a!

b)(a

#b)

"a2

#b2

A p

atte

rn a

lso

exis

ts f

or f

indi

ng t

he c

ube

of a

sum

(a!

b)3 .

1.F

ind

the

prod

uct

of (a

!b)

(a!

b)(a

!b)

.

a3!

3a2 b

!3a

b2

!b

3

2.U

se t

he p

atte

rn f

rom

Exe

rcis

e 1

to e

valu

ate

(x!

2)3 .

x3

!6x

2!

12x

!8

3.B

ased

on

your

ans

wer

to

Exe

rcis

e 1,

pred

ict

the

patt

ern

for

the

cube

of

a di

ffer

ence

(a#

b)3 .

a3"

3a2 b

!3a

b2

"b

3

4.F

ind

the

prod

uct

of (a

#b)

(a#

b)(a

#b)

and

com

pare

it t

o yo

ur

answ

er f

or E

xerc

ise

3.

a3"

3a2 b

!3a

b2

"b

3

5.U

se t

he p

atte

rn f

rom

Exe

rcis

e 4

to e

valu

ate

(x!

4)3 .

x3

!12

x2!

48x

!64

Fin

d e

ach

pro

du

ct.

6.(x

!6)

37.

(x#

10)3

x3

!18

x2!

108x

!21

6x

3"

30x2

!30

0x"

1000

8.(3

x#

y)3

9.(2

x#

y)3

27x3

"27

x2y

!9x

y2

"y

38x

3"

12x2

y!

6xy

2"

y3

10.(

4x!

3y)3

11.(

5x!

2)3

64x3

!14

4x2 y

!10

8xy

2!

27y

312

5x3

!15

0x2

!60

x!

8

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-8

8-8


Documents

Chapter 8 Resource Masters - Math Class · ©Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide to Using the Chapter 8 Resource Masters The Fast File Chapter Resource system