SECTION P.3 Radicals and Rational Exponentswps.prenhall.com/wps/media/objects/549/562624/chp_2.pdf · SECTION P.3 Radicals and Rational Exponents ... radical expressions and the use

24 • Chapter P • Prerequisites: Fundamental Concepts of Algebra

1 Evaluate square roots.

Objectives1. Evaluate square roots.

2. Use the product rule tosimplify square roots.

3. Use the quotient rule tosimplify square roots.

4. Add and subtract squareroots.

5. Rationalizedenominators.

6. Evaluate and performoperations with higherroots.

7. Understand and userational exponents.

SECTION P.3 Radicals and Rational Exponents

What is the maximum speed at which a racing cyclist can turn a corner withouttipping over? The answer, in miles per hour, is given by the algebraicexpression where x is the radius of the corner, in feet. Algebraicexpressions containing roots describe phenomena as diverse as a wild animal’sterritorial area, evaporation on a lake’s surface, and Albert Einstein’s bizarreconcept of how an astronaut moving close to the speed of light would barelyage relative to friends watching from Earth. No description of your world canbe complete without roots and radicals. In this section, we review the basics ofradical expressions and the use of rational exponents to indicate radicals.

Square RootsThe principal square root of a nonnegative real number b, written is thatnumber whose square equals b. For example,

because and because

Observe that the principal square root of a positive number is positive and theprincipal square root of 0 is 0.

The symbol that we use to denote the principal square root is called aradical sign. The number under the radical sign is called the radicand. Togetherwe refer to the radical sign and its radicand as a radical.

The following definition summarizes our discussion:

Definition of the Principal Square RootIf a is a nonnegative real number, the nonnegative number b such that

denoted by is the principal square root of a.

In the real number system, negative numbers do not have square roots. Forexample, is not a real number because there is no real number whosesquare is

If a number is nonnegative then . For example,

A22B2 = 2, A23B2 = 3, A24B2 = 4, and A25B2 = 5.

A1aB2 = a(a � 0),-9.2-9

b = 1a ,b2= a,

1

02= 0.20 = 0102

= 1002100 = 10

2b ,

41x ,

BLITMCPA.QXP.131013599_1-47 12/30/02 10:35 AM Page 24

A number that is the square of a rational number is called a perfectsquare. For example,

64 is a perfect square because

is a perfect square because

The following rule can be used to find square roots of perfect squares:

Square Roots of Perfect Squares

For example, and

The Product Rule for Square RootsA square root is simplified when its radicand has no factors other than 1 thatare perfect squares. For example, is not simplified because it can beexpressed as and is a perfect square. The product rule forsquare roots can be used to simplify

The Product Rule for Square RootsIf a and b represent nonnegative real numbers, then

and

The square root of a product is the product of the square roots.

Example 1 shows how the product rule is used to remove from the squareroot any perfect squares that occur as factors.

EXAMPLE 1 Using the Product Rule to Simplify Square Roots

Simplify: a. b.

Solutiona. 100 is the largest perfect square factor of 500.

b. We can simplify using the power rule only if 6x and 3xrepresent nonnegative real numbers.Thus,

Multiply.

9 is the largest perfect square factor of 18.

Split into two square roots.

(because 32=9) and2x2= x because x � 0.

29 = 3 = 3x22

29x2 = 29 2x2 22

2ab = 1a 2b = 29x2 22

= 29x2 � 2

= 218x2

1a 2b = 2ab 26x � 23x = 26x � 3x

x � 0.26x � 23x

2100 = 10 = 1025

2ab = 1a 2b = 2100 25 2500 = 2100 � 5

26x � 23x.2500

1a 1b = 1ab .1ab = 1a 1b

1500 .11001100 � 5

1500

2(-6)2= ∑-6∑ = 6.262

= 6

2a2= ∑a∑

19

= a 13b 2

.19

64 = 82.

Section P.3 • Radicals and Rational Exponents • 25

2 Use the product rule tosimplify square roots.


�x2 = x because x ≥ 0.

Simplify:

a. b. .

The Quotient Rule for Square RootsAnother property for square roots involves division.

The Quotient Rule for Square RootsIf a and b represent nonnegative real numbers and then

and

The square root of a quotient is the quotient of the square roots.

EXAMPLE 2 Using the Quotient Rule to Simplify Square Roots

Simplify: a. b.

Solution

a.

b. We can simplify the quotient of and using the quotient ruleonly if and 6x represent nonnegative real numbers. Thus,

Simplify: a. b.

Adding and Subtracting Square RootsTwo or more square roots can be combined provided that they have the sameradicand. Such radicals are called like radicals. For example,

EXAMPLE 3 Adding and Subtracting Like Radicals

Add or subtract as indicated:

a. b. 25x - 725x.722 + 522

7211 + 6211 = (7 + 6)211 = 13211 .

2150x322x.B25

16CheckPoint

2

248x326x= C48x3

6x= 28x2

= 24x2 22 = 24 2x2

22 = 2x 22

x � 0.48x326x248x3

B1009

=

210029=

103

248x326x.B100

9

Ba

b=

1a2b .

1a2b= Ba

b

b Z 0,

25x � 210x232

CheckPoint

1


3 Use the quotient rule tosimplify square roots.

4 Add and subtract squareroots.


Solutiona. Apply the distributive property.

Simplify.

b. Write as

Apply the distributive property.

Simplify.


a. b.

In some cases, radicals can be combined once they have been simplified.For example, to add and we can write as because 4 is aperfect square factor of 8.

EXAMPLE 4 Combining Radicals That First Require Simplification


a. b.

Solutiona.

Split 12 into two factors such that one is a perfect square.

Apply the distributive property. You will find that this stepis usually done mentally.

Simplify.

b.25 is the largest perfect square factor of 50and 16 is the largest perfect square factor of32.

and 101

Multiply.Apply the distributive property.Simplify.


a. b. 6218x - 428x.5227 + 212

CheckPoint

4

= -422x

= (20 - 24)22x = 2022x - 2422x

216 � 2 = 216 22 = 422.225 � 2 = 225 22 = 522 = 4 � 522x - 6 � 422x

= 4225 � 2x - 6216 � 2x

4250x - 6232x

= 923

= (7 + 2)2324 � 3 = 24 23 = 223 = 723 + 223

= 723 + 24 � 3723 + 212

4250x - 6232x.723 + 212

22 + 28 = 22 + 24 � 2 = 122 + 222 = (1 + 2)22 = 322

24 � 22828 ,22

217x - 20217x.8213 + 9213

CheckPoint

3

= -625x

= (1 - 7)25x

125x .25x 25x - 725x = 125x - 725x

= 1222

722 + 522 = (7 + 5)22A Radical Idea:Time Is Relative

What does travel in space have todo with radicals? Imagine that inthe future we will be able to travelat velocities approaching thespeed of light (approximately186,000 miles per second).According to Einstein’s theoryof relativity, time would passmore quickly on Earth than itwould in the moving spaceship.The expression

gives the aging rate of anastronaut relative to the agingrate of a friend on Earth, Rf. Inthe expression, v is the astronaut’sspeed and c is the speed of light.As the astronaut’s speedapproaches the speed of light,we can substitute c for v :

Let v=c.

Close to the speed of light, theastronaut’s aging rate relative toa friend on Earth is nearly 0.What does this mean? As we agehere on Earth, the space travelerwould barely get older. Thespace traveler would return to afuturistic world in which friendsand loved ones would be longdead.

= Rf 20 = 0

= Rf 21 - 12

Rf C1 - a v

cb 2

Rf C1 - a v

cb 2




Rationalizing Denominators

You can use a calculator to compare the approximate values for and

The two approximations are the same. This is not a coincidence:

This process involves rewriting a radical expression as an equivalent expressionin which the denominator no longer contains a radical. The process is calledrationalizing the denominator. If the denominator contains the square root of anatural number that is not a perfect square, multiply the numerator anddenominator by the smallest number that produces the square root of a perfectsquare in the denominator.

EXAMPLE 5 Rationalizing Denominators

Rationalize the denominator: a. b.

Solutiona. If we multiply numerator and denominator by the denominator

becomes Therefore, we multiply by 1, choosing for 1.

b. The smallest number that will produce a perfect square in the denominator

of is because We multiply by 1,

choosing for 1.

Rationalize the denominator: a. b.6212

.523

CheckPoint

5

1228=

1228�2222

=

1222216=

12224

= 322

2222

28 � 22 = 216 = 4.22 ,1228

1526=

1526�2626

=

1526236=

15266

=

5262

262626 � 26 = 236 = 6.

26 ,

1228.

1526

Any number divided by itself is 1.Multiplication by 1 does not

change the value of 1�3−−−− .

=

2329=

233

.2323

123=

123�

233

.123

5 Rationalize denominators.

Simplify: .156

15 ÷ 36 ÷ 3−− = −−−−− = −−52Multiply by 1.



How can we rationalize a denominator if the denominator contains twoterms? In general,

Notice that the product does not contain a radical. Here are some specificexamples.

The DenominatorContains: Multiply by: The New Denominator Contains:

EXAMPLE 6 Rationalizing a Denominator Containing Two Terms

Rationalize the denominator:

Solution If we multiply the numerator and denominator by the denominator will not contain a radical. Therefore, we multiply by 1, choosing

for 1.

Rationalize the denominator:8

4 + 25 .Check

Point6

In either form of theanswer, there is no

radical in the denominator.

=

7A5 - 23B22

or 35 - 723

22 .

Multiply by 1.

7

5 + 23=

7

5 + 23�

5 - 23

5 - 23=

7A5 - 23B52

- A23B2 =

7A5 - 23B25 - 3

5 - 23

5 - 23

5 - 23 ,

7

5 + 23 .

A27B2 - A23B2 = 7 - 3 = 427 - 2327 + 23

A23B2 - 62= 3 - 36 = -3323 + 623 - 6

72- A25B2 = 49 - 5 = 447 - 257 + 25

A1a + 2bB A1a - 2bB = A1aB2 - A2bB2 = a - b.



Other Kinds of RootsWe define the principal nth root of a real number a, symbolized by as follows:

Definition of the Principal nth Root of a Real Number

means that

If n, the index, is even, then a is nonnegative and b is alsononnegative If n is odd, a and b can be any real numbers.

For example,

because 43=64 and because

The same vocabulary that we learned for square roots applies to nth roots.The symbol is called a radical and a is called the radicand.

A number that is the nth power of a rational number is called a perfectnth power. For example, 8 is a perfect third power, or perfect cube, because

In general, one of the following rules can be used to find nth roots ofperfect nth powers:

Finding nth Roots of Perfect nth PowersIf n is odd,If n is even,

For example,

and

The Product and Quotient Rules for Other RootsThe product and quotient rules apply to cube roots, fourth roots, and all higherroots.

The Product and Quotient Rules for nth RootsFor all real numbers, where the indicated roots represent real numbers,

and

EXAMPLE 7 Simplifying, Multiplying,and Dividing Higher Roots

Simplify: a. b. c.

Solution

a. Find the largest perfect cube that is a factor of 24. so8 is a perfect cube and is the largest perfect cube factor of 24.

23 8 = 2, 23 24 = 23 8 � 3

B4 8116

.24 8 � 24 423 24

1n a2n b= Bn a

b, b Z 0.1n a � 2n b = 2n ab

Absolute value is not needed with oddroots, but is necessary with even roots.

24 (-2)4= ∑-2∑ = 2.23 (-2)3

= -2

2n an= ∑a∑.

2n an= a.

8 = 23.

1n a

(-2)5= -32.25 -32 = -223 64 = 4

(b � 0).(a � 0)

bn= a.1n a = b

1n a ,6 Evaluate and perform

operations with higherroots.



7 Understand and userational exponents.

b.

Find the largest perfect fourth power that is a factor of 32.

so 16 is a perfect fourth power and is thelargest perfect fourth power that is a factor of 32.

c.

because 34=81 and because 24=16.

Simplify: a. b. c.

We have seen that adding and subtracting square roots often involves simplifying terms.The same idea applies to adding and subtracting nth roots.

EXAMPLE 8 Combining Cube Roots

Subtract:

Solution

Because and 8 is the largest perfectcube that is a factor of 16.

Multiply.

Apply the distributive property.

Simplify.

Subtract:

Rational ExponentsAnimals in the wild have regions to which they confine their movement, calledtheir territorial area. Territorial area, in square miles, is related to an animal’sbody weight. If an animal weighs W pounds, its territorial area is

square miles.W to the what power?! How can we interpret the information given by this

algebraic expression?

W141�100

323 81 - 423 3 .CheckPoint

8

= -123 2 or - 23 2

= (10 - 11)23 2

= 1023 2 - 1123 2

23 8 � 2 = 23 8 23 2 = 223 2= 5 � 223 2 - 1123 2

23 8 = 2,16 = 8�2= 523 8 � 2 - 1123 2

523 16 - 1123 2

523 16 - 1123 2 .

B3 12527

.25 8 � 25 823 40CheckPoint

7

24 16 = 224 81 = 3 =

32

Bn ab

=

1n a2n b B4 81

16=

24 8124 16

= 224 2

2n ab = 1n a � 2n b = 24 16 � 24 2

24 16 = 2, = 24 16 � 2

= 24 32

1n a � 2n b = 2n ab 24 8 � 24 4 = 24 8 � 4

= 223 3

2n ab = 1n a 2n b = 23 8 � 23 3

Some higher even and oddroots occur so frequentlythat you might want tomemorize them.

Cube Roots

Fourth FifthRoots Roots

24 625 = 524 256 = 4

52243 = 324 81 = 3

5232 = 224 16 = 2

521 = 124 1 = 1

23 64 = 423 1000 = 1023 27 = 323 216 = 623 8 = 223 125 = 523 1 = 1

Study Tip



In the last part of this section, we turn our attention to rational exponentssuch as and their relationship to roots of real numbers.

Definition of Rational ExponentsIf represents a real number and is an integer, then

Furthermore,

EXAMPLE 9 Using the Definition of

Simplify: a. b. c.

Solution

a. b.

c.

Simplify: a. b. c.

Note that every rational exponent in Example 9 has a numerator of 1 orWe now define rational exponents with any integer in the numerator.

Definition of Rational Exponents

If represents a real number, is a rational number reduced to lowest

terms, and is an integer, then

The exponent m�n consists of two parts: the denominator n is the root andthe numerator m is the exponent. Furthermore,

EXAMPLE 10 Using the Definition of

Simplify: a. b. c. 16-3�4.93�2272�3

am/n

a-m�n=

1

am�n .

am�n= A1n aBm = 2n am

.

n � 2

mn

1n a

-1.

32-1�5.271�3811�2CheckPoint

9

64-1�3=

1

641�3 =

123 64=

14

81�3= 23 8 = 2641�2

= 264 = 8

64-1�3.81�3641�2

a1/n

a-1�n=

1

a1�n=

11n a , a Z 0.

a1�n= 1n a .

n � 21n a

141100



Solution

a.

b.

c.

Simplify: a. b.

Properties of exponents can be applied to expressions containing rational exponents.

EXAMPLE 11 Simplifying Expressions with Rational Exponents

Simplify using properties of exponents:

a. b.

Solution

a. Group factors with the same base.

When multiplying expressions with the samebase, add the exponents.

b. Group factors with the same base.

When dividing expressions with the same base, subtract the exponents.

Simplify: a. b.20x4

5x3�2 .(2x4�3)(5x8�3)CheckPoint

11

53 -

34 =

2012 -

912 =

1112 = 2x11�12

= 2x(5�3) - (3�4)

32x5�3

16x3�4 = a 3216b a x5�3

x3�4 b

12 +

34 =

24 +

34 =

54= 35x5�4

= 35x(1�2) + (3�4)

(5x1�2)(7x3�4) = 5 � 7x1�2 � x3�4

32x5�3

16x3�4 .(5x1�2)(7x3�4)

32-2�5.43�2CheckPoint

10

16-3�4=

1

163�4 =

1

A24 16B3 =

123 =

18

93�2= A29B3 = 33

= 27

272�3= A23 27B2 = 32

= 9Here are the calculatorkeystroke sequences for 272/3:

Many Scientific Calculators

Many Graphing Calculators

ENTER.27 ¿ ( 2 , 3 )

27 yx ( 2 , 3 ) =

Technology

The denominator of is the root and

the numerator is the exponent.

23−−


38.

In Exercises 39–48, rationalize the denominator.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

Evaluate each expression in Exercises 49–60, or indicate thatthe root is not a real number.

49. 50.

51. 52.

53. 54.

55. 56.

57. 58.

59. 60.

Simplify the radical expressions in Exercises 61–68.

61. 62.

63. 64.

65. 66.

67. 68.

In Exercises 69–76, add or subtract terms whenever possible.

69. 70.

71. 72.

73. 74.

75. 76. 23 + 23 1522 + 23 8

23 24xy3- y23 81x23 54xy3

- y23 128x

323 24 + 23 81523 16 + 23 54

625 3 + 225 3425 2 + 325 2

24 162x524 2x

25 64x625 2x

23 12 � 23 423 9 � 23 6

23 x523 x4

23 15023 32

26 16425 -

132

25 (-2)525 (-3)5

24 (-2)424 (-3)4

24 -8124 -16

23 -12523 -8

23 823 125

1127 - 23

625 + 23

523 - 1

725 - 2

3

3 + 27

13

3 + 211

2723

2225

2210

127

3254 - 2224 - 296 + 4263


Practice ExercisesEvaluate each expression in Exercises 1–6 or indicate that the root is not a real number.

1. 2.3. 4.5. 6.

Use the product rule to simplify the expressions in Exercises 7 16. In Exercises 11 16, assume that variablesrepresent nonnegative real numbers.

7. 8.9. 10.

11. 12.

13. 14.

15. 16.

Use the quotient rule to simplify the expressions in Exercises17–26. Assume that

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

In Exercises 27–38, add or subtract terms whenever possible.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 328 - 232 + 3272 - 275

4212 - 22753218 + 5250

263x - 228x250x - 28x

220 + 62528 + 322

4213x - 6213x6217x - 8217x

825 + 1125723 + 623

2500x3210x-1

2200x3210x-1

224x423x

2150x423x

272x328x

248x323x

B1219B49

16

B 149B 1

81

x 7 0.

26x � 23x222x2 � 26x

2y32x3

210x � 28x22x � 26x

2125x2245x2

227250

--

2(-17)22(-13)2

2-252-36225236

Rational exponents are sometimes useful for simplifying radicals by reducing their index.

EXAMPLE 12 Reducing the Index of a Radical

Simplify:

Solution

Simplify: 26 x3 .Check

Point12

29 x3= x3�9

= x1�3= 23 x

29 x3 .

EXERCISE SET P.3


Exercise Set P.3 • 35

In Exercises 77–84, evaluate each expression without using acalculator.

77. 78.79. 80.81. 82.83. 84.

In Exercises 85–94, simplify using properties of exponents.

85. 86.

87. 88.

89. 90.

91. 92.

93. 94.

In Exercises 95–102, simplify by reducing the index of theradical.

95. 96.

97. 98.

99. 100.

101. 102.

Application Exercises

103. The algebraic expression is used to estimate thespeed of a car prior to an accident, in miles per hour,based on the length of its skid marks, L, in feet. Findthe speed of a car that left skid marks 40 feet long, andwrite the answer in simplified radical form.

104. The time, in seconds, that it takes an object to fall adistance d, in feet, is given by the algebraic expression

Find how long it will take a ball dropped from the

top of a building 320 feet tall to hit the ground. Writethe answer in simplified radical form.

105. The early Greeks believed that the most pleasing of allrectangles were golden rectangles whose ratio of widthto height is

Rationalize the denominator for this ratio and then usea calculator to approximate the answer correct to thenearest hundredth.

w

h=

225 - 1 .

B d

16 .

225L

122x4y829 x6y3

29 x626 x4

24 x1223 x6

24 7224 52

(2y1�5)4

y3�10

(3y1�4)3

y1�12

(125x9y6)1�3(25x4y6)1�2

(x4�5)5(x2�3)3

72x3�4

9x1�3

20x1�2

5x1�4

(3x2�3)(4x3�4)(7x1�3)(2x1�4)

16-5�232-4�5

82�31252�3

271�381�3

1211�2361�2

106. The amount of evaporation, in inches per day, of alarge body of water can be described by the algebraicexpression

where

surface area of the water, in square milesaverage wind speed of the air over the water,in miles per hour.

Determine the evaporation on a lake whose surface areais 9 square miles on a day when the wind speed over thewater is 10 miles per hour.

107. In the Peanuts cartoon shown below,Woodstock appearsto be working steps mentally. Fill in the missing steps that

show how to go from to

PEANUTS reprinted by permission of United Feature Syndicate, Inc.

108. The algebraic expression describes thepercentage of U.S. taxpayers who are a years old whofile early. Evaluate the algebraic expression for Describe what the answer means in practical terms.

109. The algebraic expression describes theduration of a storm, in hours, whose diameter is dmiles. Evaluate the algebraic expression for Describe what the answer means in practical terms.

Writing in Mathematics

110. Explain how to simplify

111. Explain how to add

112. Describe what it means to rationalize a denominator.

Use both and in your explanation.1

5 + 25

125

23 + 212 .

210 � 25 .

d = 9.

0.07d3�2

a = 32.

152a-1�5

73

23 .722 � 2 � 3

6

w =

a =

w

201a



Objectives1. Understand the

vocabulary ofpolynomials.

2. Add and subtractpolynomials.

3. Multiply polynomials.

4. Use FOIL in polynomialmultiplication.

5. Use special products inpolynomialmultiplication.

6. Perform operations withpolynomials in severalvariables.

SECTION P.4 Polynomials

113. What difference is there in simplifying and

114. What does mean?115. Describe the kinds of numbers that have rational fifth

roots.116. Why must a and b represent nonnegative numbers

when we write Is it necessary touse this restriction in the case of Explain.

Technology Exercises

117. The algebraic expression

describes the percentage of people in the United States applying for jobs t years after 1985 who tested positive for illegal drugs. Use a calculator to find the percentage whotested positive from 1986 through 2001.Round answers tothe nearest hundredth of a percent. What trend do you observe for the percentage of potential employees testingpositive for illegal drugs over time?

118. The territorial area of an animal in the wild is defined tobe the area of the region to which the animal confines itsmovements. The algebraic expression describes theterritorial area, in square miles, of an animal that weighs

W1.41

73t1�3- 28t2�3

t

13 a � 23 b = 23 ab ?1a � 2b = 2ab ?

am�n

24 (-5)4 ?

23 (-5)3 W pounds. Use a calculator to find the territorial area of animals weighing 25, 50, 150, 200, 250,and 300 pounds. What do the values indicate about therelationship between body weight and territorial area?

Critical Thinking Exercises

119. Which one of the following is true?

a. Neither nor represent real numbers.b.c.d.

In Exercises 120 121, fill in each box to make the statementtrue.

120.

121.

122. Find exact value of without

the use of a calculator.

123. Place the correct symbol, , in the box betweeneach of the given numbers. Do not use a calculator.Then check your result with a calculator.a. b. 27 + 218 n 27 + 1831�2

n 31�3

7 or 6

B13 + 22 +

7

3 + 22

2nxn= 5x7

A5 + 2nB A5 - 2nB = 22

-

21�2 � 21�2= 2

8-1�3= -2

2x2+ y2

= x + y

(-8)1�3(-8)1�2

This computer-simulatedmodel of the common coldvirus was developed byresearchers at PurdueUniversity. Their discoveryof how the virus infectshuman cells could lead tomore effective treatmentfor the illness.


We can express 0 in manyways, including 0x, 0x2, and0x3. It is impossible to assign asingle exponent on thevariable. This is why 0 has nodefined degree.

Study Tip

Section P.4 • Polynomials • 37

1 Understand the vocabularyof polynomials.

Runny nose? Sneezing? You are probably familiar with the unpleasant onsetof a cold. We “catch cold” when the cold virus enters our bodies, where itmultiplies. Fortunately, at a certain point the virus begins to die. The algebraicexpression describes the billions of viral particles in ourbodies after x days of invasion. The expression enables mathematicians todetermine the day on which there is a maximum number of viral particles and,consequently, the day we feel sickest.

The algebraic expression is an example of a polynomial.A polynomial is a single term or the sum of two or more terms containing variableswith whole number exponents. This particular polynomial contains three terms.Equations containing polynomials are used in such diverse areas as science,business, medicine, psychology, and sociology. In this section, we review basic ideasabout polynomials and their operations.

The Vocabulary of PolynomialsConsider the polynomial

We can express this polynomial as

The polynomial contains four terms. It is customary to write the terms in theorder of descending powers of the variable. This is the standard form of apolynomial.

We begin this section by limiting our discussion to polynomials containingonly one variable. Each term of a polynomial in x is of the form The degreeof is n. For example, the degree of the term is 3.

The Degree of If the degree of is n. The degree of a nonzero constant is 0. Theconstant 0 has no defined degree.

Here is an example of a polynomial and the degree of each of its four terms:

Notice that the exponent on x for the term 2x is understood to be For thisreason, the degree of 2x is 1. You can think of as thus, its degree is 0.

A polynomial which when simplified has exactly one term is called amonomial. A binomial is a simplified polynomial that has two terms, each witha different exponent. A trinomial is a simplified polynomial with three terms,each with a different exponent. Simplified polynomials with four or more termshave no special names.

The degree of a polynomial is the highest degree of all the terms of thepolynomial. For example, is a binomial of degree 2 because thedegree of the first term is 2, and the degree of the other term is less than 2.Also, is a trinomial of degree 5 because the degree of the firstterm is 5, and the degrees of the other terms are less than 5.

7x5- 2x2

+ 4

4x2+ 3x

-5x0;-51: 2x1.

6x4- 3x3

+ 2x - 5.

axna Z 0,

axn

7x3axnaxn.

7x3+ (-9x2) + 13x + (-6).

7x3- 9x2

+ 13x - 6.

-0.75x4+ 3x3

+ 5

-0.75x4+ 3x3

+ 5

degree4

degree3

degree1

degree of non-zero constant: 0



Up to now, we have used x to represent the variable in a polynomial.However, any letter can be used. For example,

• is a polynomial (in x) of degree 5.• is a polynomial (in y) of degree 3.• is a polynomial (in z) of degree 7.

Not every algebraic expression is a polynomial. Algebraic expressionswhose variables do not contain whole number exponents such as

are not polynomials. Furthermore, a quotient of polynomials such as

is not a polynomial because the form of a polynomial involves only addition andsubtraction of terms, not division.

We can tie together the threads of our discussion with the formaldefinition of a polynomial in one variable. In this definition, the coefficients ofthe terms are represented by (read “a sub n”), (read “a sub n minus 1”),

and so on.The small letters to the lower right of each a are called subscriptsand are not exponents. Subscripts are used to distinguish one constant fromanother when a large and undetermined number of such constants are needed.

Definition of a Polynomial in xA polynomial in x is an algebraic expression of the form

where are real numbers, and n is anonnegative integer. The polynomial is of degree n, is the leadingcoefficient, and is the constant term.

Adding and Subtracting PolynomialsPolynomials are added and subtracted by combining like terms. For example,we can combine the monomials and using addition as follows:

EXAMPLE 1 Adding and Subtracting Polynomials

Perform the indicated operations and simplify:

a.b.

Solution

a.

Group like terms.

Combine like terms.

Simplify.= 4x3+ 9x2

- 13x - 3

= 4x3+ 9x2

+ (-13x) + (-3)

+ (-5x - 8x) + (3 - 6)

= (-9x3+ 13x3) + (7x2

+ 2x2)

(-9x3+ 7x2

- 5x + 3) + (13x3+ 2x2

- 8x - 6)

(7x3- 8x2

+ 9x - 6) - (2x3- 6x2

- 3x + 9).(-9x3

+ 7x2- 5x + 3) + (13x3

+ 2x2- 8x - 6)

-9x3+ 13x3

= (-9 + 13)x3= 4x3.

13x3-9x3

a0

an

an Z 0,an, an-1, an-2, p , a1, and a0

anxn+ an-1x

n-1+ an-2x

n-2+

p+ a1x + a0,

an-2,an-1an

x2+ 2x + 5

x3- 7x2

+ 9x - 3

3x-2+ 7 and 5x3�2

+ 9x1�2+ 2

z7+ 12

6y3+ 4y2

- y + 37x5

- 3x3+ 8

2 Add and subtract polynomials.


b.Rewrite subtractionas addition of the additive inverse. Besure to change thesign of each term inside parenthesespreceded by the negative sign.Group like terms.

Combine like terms.Simplify.

Perform the indicated operations and simplify:

a.b.

Multiplying PolynomialsThe product of two monomials is obtained by using properties of exponents.For example,

Furthermore, we can use the distributive property to multiply a monomial and apolynomial that is not a monomial. For example,

How do we multiply two polynomials if neither is a monomial? Forexample, consider

One way to perform this multiplication is to distribute 2x throughout thetrinomial

and 3 throughout the trinomial

Then combine the like terms that result.

Multiplying Polynomials when Neither is a MonomialMultiply each term of one polynomial by each term of the otherpolynomial. Then combine like terms.

3(x2+ 4x + 5).

2x(x2+ 4x + 5)

binomial trinomial

(2x + 3)(x2+ 4x + 5).

3x4(2x3- 7x + 3) = 3x4 � 2x3

- 3x4 � 7x + 3x4 � 3 = 6x7- 21x5

+ 9x4.

(-8x6)(5x3) = -8 � 5x6 + 3= -40x9.

(13x3- 9x2

- 7x + 1) - (-7x3+ 2x2

- 5x + 9).(-17x3

+ 4x2- 11x - 5) + (16x3

- 3x2+ 3x - 15)

CheckPoint

1

= 5x3- 2x2

+ 12x - 15= 5x3

+ (-2x2) + 12x + (-15)+ (9x + 3x) + (-6 - 9)

= (7x3- 2x3) + (-8x2

+ 6x2)

= (7x3- 8x2

+ 9x - 6) + (-2x3+ 6x2

+ 3x - 9)(7x3

- 8x2+ 9x - 6) - (2x3

- 6x2- 3x + 9)


3 Multiply polynomials.

You can also arrange liketerms in columns and combinevertically:

The like terms can becombined by adding theircoefficients and keeping thesame variable factor.

7x3- 8x2

+ 9x - 6-2x3

+ 6x2+ 3x - 9

5x3- 2x2

+ 12x - 15

Study Tip

Multiply coefficients and add exponents.

monomial trinomial


F

outside

first last

inside

O I L


EXAMPLE 2 Multiplying a Binomial and a Trinomial

Multiply:

Solution

Multiply the trinomialby each term of thebinomial.Use the distributiveproperty.

Multiply the monomials: multiplycoefficients and addexponents.

Combine like terms:

Another method for solving Example 2 is to use a vertical format similarto that used for multiplying whole numbers.

Multiply:

The Product of Two Binomials: FOILFrequently we need to find the product of two binomials. We can use a methodcalled FOIL, which is based on the distributive property, to do so. For example,we can find the product of the binomials as follows:

First, distribute 3x overThen distribute 2.

Two binomials can be quickly multiplied by using the FOIL method, in whichF represents the product of the first terms in each binomial, O represents theproduct of the outside terms, I represents the product of the two inside terms, and Lrepresents the product of the last, or second, terms in each binomial.

Combine like terms.= 12x2+ 23x + 10

(3x + 2)(4x + 5) = 12x2+ 15x + 8x + 10

= 12x2+ 15x + 8x + 10.

= 3x(4x) + 3x(5) + 2(4x) + 2(5)

4x + 5.(3x + 2)(4x + 5) = 3x(4x + 5) + 2(4x + 5)

3x + 2 and 4x + 5

(5x - 2)(3x2- 5x + 4).

CheckPoint

2

x2+ 4x + 5

2x + 33x2

+ 12x + 152x3

+ 8x2+ 10x

2x3+ 11x2

+ 22x + 15

10x + 12x = 22x.8x2

+ 3x2= 11x2 and

= 2x3+ 11x2

+ 22x + 15

= 2x3+ 8x2

+ 10x + 3x2+ 12x + 15

= 2x � x2+ 2x � 4x + 2x � 5 + 3 � x2

+ 3 � 4x + 3 � 5

= 2x(x2+ 4x + 5) + 3(x2

+ 4x + 5)(2x + 3)(x2

+ 4x + 5)

(2x + 3)(x2+ 4x + 5).

4 Use FOIL in polynomialmultiplication.

Write like terms inthe same column.

Combine like terms.

3(x2 + 4x + 5)

2x(x2 + 4x + 5)



5 Use special products inpolynomial multiplication.

In general, here is how to use the FOIL method to find the product of

Using the FOIL Method to Multiply Binomials

EXAMPLE 3 Using the FOIL Method

Multiply:

Solution

Combine like terms.

Multiply:

Multiplying the Sum and Difference of Two TermsWe can use the FOIL method to multiply as follows:

.

Notice that the outside and inside products have a sum of 0 and the termscancel. The FOIL multiplication provides us with a quick rule for multiplyingthe sum and difference of two terms, referred to as a special-product formula.

The Product of the Sum and Difference of Two Terms

The product of the sumand the difference of the

same two terms

the square of the firstterm minus the square of

the second term.

is

(A + B)(A - B) = A2- B2

(A + B)(A - B) = A2- AB + AB - B2

= A2- B2.

F O I L

A + B and A - B

(7x - 5)(4x - 3).CheckPoint

3

= 15x2+ 11x - 12

= 15x2- 9x + 20x - 12

(3x + 4)(5x - 3) = 3x � 5x + 3x(-3) + 4 � 5x + 4(-3)

(3x + 4)(5x - 3).

(ax + b)(cx + d) = ax � cx + ax � d + b � cx + b � d

ax + b and cx + d:

outside

first last

inside

F O I L

Product ofFirst terms

Product ofOutside terms

Product ofInside terms

Product ofLast termsoutside

first last

inside



EXAMPLE 4 Finding the Product of the Sum and Difference of Two Terms

Find each product:

a. b.

Solution Use the special-product formula shown.

a.

b.

Find each product:

a. b.

The Square of a BinomialLet us find , the square of a binomial sum. To do so, we begin withthe FOIL method and look for a general rule.

This result implies the following rule, which is another example of a special-product formula:

The Square of a Binomial Sum

EXAMPLE 5 Finding the Square of a Binomial Sum

Square each binomial:

a. b. (3x + 7)2.(x + 3)2

The square of abinomial sum

first termsquared

2 times theproduct ofthe terms

lastterm

squared.

is plus plus

B2+2AB+A2

=(A + B)2

= A2+ 2AB + B2

(A + B)2= (A + B)(A + B) = A � A + A � B + A � B + B � B

F O I L

(A + B)2

(2y3- 5)(2y3

+ 5).(7x + 8)(7x - 8)

CheckPoint

4

= 25a8- 3662

-(5a4)2=(5a4

+ 6)(5a4- 6)

= 16y2- 932

-(4y)2=(4y + 3)(4y - 3)

First termsquared

Second termsquared− = Product

(A + B)(A - B) = A2- B2

(5a4+ 6)(5a4

- 6).(4y + 3)(4y - 3)

Incorrect

The middle term2AB is missing

Caution! The square of a sumis not the sum of the squares.

Show that andare not equal by

substituting 5 for x in eachexpression and simplifying.

x2+ 9

(x + 3)2

(x + 3)2Z x2

+ 9

(A + B)2Z A2

+ B2

Study Tip



Square aDifference – of the Terms ±

a.

b. = 25y2- 60y + 3662

+2(5y)(6)-(5y)2(5y - 6)2=

= x2- 8x + 1642

+2 � x � 4-x2(x - 4)2=

= Product(Last Term)2Term)22 � Product(First


B2+2AB+A2(A + B)2

=

Squarea Sum ± of the Terms ±

a.

b. = 9x2+ 42x + 4972

+2(3x)(7)+(3x)2(3x + 7)2=

= x2+ 6x + 932

+2 � x � 3+x2(x + 3)2=

= Product(Last Term)2Term)22 � Product(First


a. b.

Using the FOIL method on , the square of a binomial difference,we obtain the following rule:

The Square of a Binomial Difference

EXAMPLE 6 Finding the Square of a Binomial Difference


a. b.


B2+2AB-A2(A - B)2

=

(5y - 6)2.(x - 4)2

The square ofa binomialdifference

first termsquared

2 times theproduct ofthe terms

lastterm

squared.

is minus plus

B2+2AB-A2

=(A - B)2

(A - B)2

(5x + 4)2.(x + 10)2

CheckPoint

5


a. b. (7x - 3)2.(x - 9)2

CheckPoint

6



Special ProductsThere are several products that occur so frequently that it’s convenient tomemorize the form, or pattern, of these formulas.

Special ProductsLet A and B represent real numbers, variables, or algebraic expressions.

Special Product Example

Sum and Difference of Two Terms

Squaring a Binomial

Cubing a Binomial

Polynomials in Several VariablesThe next time you visit the lumber yard and go rummaging through piles ofwood, think polynomials, although polynomials a bit different from those wehave encountered so far. The construction industry uses a polynomial in twovariables to determine the number of board feet that can be manufacturedfrom a tree with a diameter of x inches and a length of y feet. This polynomialis

In general, a polynomial in two variables, x and y, contains the sum of oneor more monomials in the form The constant, a, is the coefficient. Theexponents, n and m, represent whole numbers. The degree of the monomial

is We’ll use the polynomial from the construction industry toillustrate these ideas.

The degree of a polynomial in two variables is the highest degree of all itsterms. For the preceding polynomial, the degree is 3.

Degree ofmonomial:2 + 1 = 3



+4y-2xy14 x2y

The coefficients are , −2, and 4.14−−

n + m.axnym

axnym.

14 x2y - 2xy + 4y.

= x3- 6x2

+ 12x - 8= x3

- 3x2(2) + 3x(2)2- 23

(x - 2)3(A - B)3= A3

- 3A2B + 3AB2- B3

= x3+ 12x2

+ 48x + 64= x3

+ 3x2(4) + 3x(4)2+ 43

(x + 4)3(A + B)3= A3

+ 3A2B + 3AB2+ B3

= 9x2- 24x + 16

= (3x)2- 2 � 3x � 4 + 42

(3x - 4)4(A - B)2= A2

- 2AB + B2= y2

+ 10y + 25(y + 5)2

= y2+ 2 � y � 5 + 52(A + B)2

= A2+ 2AB + B2

= 4x2- 9

(2x + 3)(2x - 3) = (2x)2- 32(A + B)(A - B) = A2

- B2

Although it’s convenient tomemorize these forms, theFOIL method can be used onall five examples in the box.Tocube you can firstsquare using FOIL andthen multiply this result by

In short, you do notnecessarily have to utilizethese special formulas.What isthe advantage of knowing andusing these forms?

x + 4.

x + 4x + 4,

Study Tip

6 Perform operations withpolynomials in severalvariables.



Polynomials containing two or more variables can be added, subtracted,and multiplied just like polynomials that contain only one variable.

EXAMPLE 7 Subtracting Polynomials in Two Variables

Subtract as indicated:

Solution

Change the sign of each term in the second polynomialand add the two polynomials.

Group like terms.

Combine like terms by combining coefficients and keepingthe same variable factors.

Subtract:

EXAMPLE 8 Multiplying Polynomials in Two Variables

Multiply: a. b.

Solution We will perform the multiplication in part (a) using the FOIL method.We will multiply in part (b) using the formula for the square of a binomial sum,

a. Multiply these binomials using the FOIL method.

Combine like terms.

b.

Multiply:

a. b. (x2+ 5y)2.(7x - 6y)(3x - y)

CheckPoint

8

= 25x2+ 30xy + 9y2

(5x + 3y)2= (5x)2

+ 2(5x)(3y) + (3y)2

(A + B)2 = A2 + 2 � A � B + B 2

= 3x2+ 7xy - 20y2

= 3x2- 5xy + 12xy - 20y2

= (x)(3x) + (x)(-5y) + (4y)(3x) + (4y)(-5y)

F O I L

(x + 4y)(3x - 5y)

(A + B)2.

(5x + 3y)2.(x + 4y)(3x - 5y)

(x3- 4x2y + 5xy2

- y3) - (x3- 6x2y + y3).Check

Point7

= 2x3- 3x2y + 5xy2

- 7

= (5x3- 3x3) + (-9x2y + 6x2y) + (3xy2

+ 2xy2) + (-4 - 3)

= (5x3- 9x2y + 3xy2

- 4) + (-3x3+ 6x2y + 2xy2

- 3)

(5x3- 9x2y + 3xy2

- 4) - (3x3- 6x2y - 2xy2

+ 3)

(5x3- 9x2y + 3xy2

- 4) - (3x3- 6x2y - 2xy2

+ 3).


Practice ExercisesIn Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in

standard form.

1. 2.

3. 4.

In Exercises 5–8, find the degree of the polynomial.

5. 6.7.8.

In Exercises 9–14, perform the indicated operations. Write theresulting polynomial in standard form and indicate its degree.

9.10.11.12.13.14.

In Exercises 15–58, find each product.

15. 16.17. 18.19. 20.21. 22.23. 24.25. 26.27. 28.29. 30.31. 32.33. 34.35. 36.37. 38.39. 40.41. 42.43. 44.45. 46.47. 48. (5x2

- 3)2(4x2- 1)2

(x - 4)2(x - 3)2

(3x + 2)2(2x + 3)2

(x + 5)2(x + 2)2

(2 - y5)(2 + y5)(1 - y5)(1 + y5)

(3x2+ 4x)(3x2

- 4x)(4x2+ 5x)(4x2

- 5x)

(4 - 3x)(4 + 3x)(5 - 7x)(5 + 7x)

(2x + 5)(2x - 5)(3x + 2)(3x - 2)

(x + 5)(x - 5)(x + 3)(x - 3)

(7x3+ 5)(x2

- 2)(8x3+ 3)(x2

- 5)

(7x2- 2)(3x2

- 5)(5x2- 4)(3x2

- 7)

(2x - 5)(7x + 2)(2x - 3)(5x + 3)

(7x + 4)(3x + 1)(3x + 5)(2x + 1)

(x - 1)(x + 2)(x - 5)(x + 3)

(x + 8)(x + 5)(x + 7)(x + 3)

(2x - 1)(x2- 4x + 3)(2x - 3)(x2

- 3x + 5)

(x + 5)(x2- 5x + 25)(x + 1)(x2

- x + 1)

(8x2+ 7x - 5) - (3x2

- 4x) - (-6x3- 5x2

+ 3)

(5x2- 7x - 8) + (2x2

- 3x + 7) - (x2- 4x - 3)

(18x4- 2x3

- 7x + 8) - (9x4- 6x3

- 5x + 7)

(17x3- 5x2

+ 4x - 3) - (5x3- 9x2

- 8x + 11)

(-7x3+ 6x2

- 11x + 13) + (19x3- 11x2

+ 7x - 17)

(-6x3+ 5x2

- 8x + 9) + (17x3+ 2x2

- 4x - 13)

x2- 8x3

+ 15x4+ 91

x2- 4x3

+ 9x - 12x4+ 63

-4x3+ 7x2

- 113x2- 5x + 4

x2- x3

+ x4- 5

2x + 3x

2x + 3x-1- 52x + 3x2

- 5


EXERCISE SET P.4

49. 50.51. 52.53. 54.55. 56.57. 58.

In Exercises 59–66, perform the indicated operations. Indicatethe degree of the resulting polynomial.

59.60.61.62.63.64.65.66.

In Exercises 67–82, find each product.

67. 68.69. 70.71. 72.73. 74.75. 76.77. 78.79. 80.81.82.

Application Exercises

83. The polynomial describes theamount, in thousands of dollars, that a person earning xthousand dollars a year feels underpaid. Evaluate thepolynomial for Describe what the answer meansin practical terms.

84. The polynomial describes thedeath rate per year, per 100,000 men, for men averagingx hours of sleep each night. Evaluate the polynomial for

Describe what the answer means in practicalterms.x = 10.

104.5x2- 1501.5x + 6016

x = 40.

0.018x2- 0.757x + 9.047

(3xy2- 4y)(3xy2

+ 4y)

(7xy2- 10y)(7xy2

+ 10y)

(7x + 3y)(7x - 3y)(3x + 5y)(3x - 5y)

(x + y)(x2- xy + y2)(x - y)(x2

+ xy + y2)

(x2y2- 5)2(x2y2

- 3)2

(9x + 7y)2(7x + 5y)2

(7x2y + 1)(2x2y - 3)(3xy - 1)(5xy + 2)

(3x - y)(2x + 5y)(x - 3y)(2x + 7y)

(x + 9y)(6x + 7y)(x + 5y)(7x + 3y)

(5x4y2+ 6x3y - 7y) - (3x4y2

- 5x3y - 6y + 8x)

(3x4y2+ 5x3y - 3y) - (2x4y2

- 3x3y - 4y + 6x)

(x4- 7xy - 5y3) - (6x4

- 3xy + 4y3)

(x3+ 7xy - 5y2) - (6x3

- xy + 4y2)

(7x4y2- 5x2y2

+ 3xy) + (-18x4y2- 6x2y2

- xy)

(4x2y + 8xy + 11) + (-2x2y + 5xy + 2)

(-2x2y + xy) + (4x2y + 7xy)

(5x2y - 3xy) + (2x2y - xy)

(2x - 3)3(3x - 4)3

(x - 1)3(x - 3)3

(3x + 4)3(2x + 3)3

(x + 2)3(x + 1)3

(9 - 5x)2(7 - 2x)2


85. The polynomial describes thenumber of violent crimes in the United States, per 100,000inhabitants, x years after 1975. Evaluate the polynomialfor Describe what the answer means in practicalterms. How well does the polynomial describe the crimerate for the appropriate year shown in the bar graph?

86. The polynomial is used by coachesto get athletes fired up so that they can perform well.Thepolynomial represents the performance level related tovarious levels of enthusiasm, from (almost noenthusiasm) to (maximum level of enthusiasm).Evaluate the polynomial for and

Describe what happens to performance as weget more and more fired up.

87. The number of people who catch a cold t weeks afterJanuary 1 is The number of people whorecover t weeks after January 1 is Write apolynomial in standard form for the number of peoplewho are still ill with a cold t weeks after January 1.

88. The weekly cost, in thousands of dollars, for producing xstereo headphones is The weekly revenue, inthousands of dollars, for selling x stereo headphones is

Write a polynomial in standard form for theweekly profit, in thousands of dollars, for producing andselling x stereo headphones.

90x2- x.

30x + 50.

t - t2+

13 t3.

5t - 3t2+ t3.

A = 80.A = 20, A = 50,

A = 100A = 1

-0.02A2+ 2A + 22

Vio

lent

Cri

mes

per

100,

000

Inha

bita

nts

Violent Crime in the United States

Year

1975 1980 1990 1995 1998 2000

150

0

300

450

600

750

487.8

596.6

731.8684.6

567.5524.7

Source: F.B.I.

x = 25.

-1.45x2+ 38.52x + 470.78

Exercise Set P.4 • 47

In Exercises 89–90, write a polynomial in standard form thatrepresents the area of the shaded region of each figure.

89.

90.

Writing in Mathematics

91. What is a polynomial in x?92. Explain how to subtract polynomials.93. Explain how to multiply two binomials using the FOIL

method. Give an example with your explanation.94. Explain how to find the product of the sum and difference

of two terms. Give an example with your explanation.95. Explain how to square a binomial difference. Give an

example with your explanation.96. Explain how to find the degree of a polynomial in two

variables.97. For Exercise 86, explain why performance levels do what

they do as we get more and more fired up. If possible,describe an example of a time when you were tooenthused and thus did poorly at something you werehoping to do well.

x + 2

x + 3 x + 1

x + 5

x + 9

x + 1

x + 4

x + 3


Documents

SECTION P.3 Radicals and Rational Exponentswps.prenhall.com/wps/media/objects/549/562624/chp_2.pdf · SECTION P.3 Radicals and Rational Exponents ... radical expressions and the use