Chapter 5 Exponents and Polynomials

CHAPTER

5

For use by Palm Beach State College only.

Exponents and Polynomials

5.1 Exponents

5.2 Adding and Subtracting

Polynomials

5.3 Multiplying Polynomials

5.4 Special Products

Integrated Review--Exponents and Operations on Polynomials

5.5 Negative Exponents and

Scientific Notation

5.6 Dividing Polynomials

CHECK YOUR PROGRESS

Vocabulary Check Chapter Highlights Chapter Review Getting Ready for the Test Chapter Test Cumulative Review

Recall from Chapter 1 that an exponent is a shorthand notation for repeated factors. This chapter explores additional concepts about exponents and exponential expressions. An especially useful type of exponential expression is a polynomial. Polynomials model many realworld phenomena. In this chapter, we focus on polynomials and operations on polynomials.

Can You Imagine a World Without the Internet? In 1995, less than 1% of the world population was connected to the Internet. By 2015, that number had increased to 40%. Technology changes so fast that, if this trend continues, by the time you read this, far more than 40% of the world population will be connected to the Internet. The circle graph below shows Internet users by region of the world in 2015. In Section 5.2, Exercises 103 and 104, we explore more about the growth of Internet users.

Worldwide Internet Users 3500

Worldwide Internet Users

3000

(by region of the world)

2500 Oceania

Europe

2000

Africa

Asia 1500 1000

Number of Internet Users (in milions)

500

Americas

0 1995 2000 2005 2010 2015 Year

Data from International Telecommunication Union and United Nations Population Division

310

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For use by Palm Beach State College only.

Section 5.1 Exponents 311

5.1 Exponents

OBJECTIVES 1 Evaluate Exponential

Expressions.

2 Use the Product Rule for

Exponents.

3 Use the Power Rule for

Exponents.

4 Use the Power Rules for

Products and Quotients.

5 Use the Quotient Rule for

Exponents, and Define a Number Raised to the 0 Power.

6 Decide Which Rule(s) to Use to

Simplify an Expression.

OBJECTIVE

1 Evaluating Exponential Expressions

AFosrweexarmevpielew, e2d#

2in#

S2e#c2ti#o2nc1a.4n,

an be

exponent is a written as 25.

shorthand notation for repeated factors. The expression 25 is called an exponen-

tial expression. It is also called the fifth power of 2, or we say that 2 is raised to the fifth

power.

56 = 5 # 5 # 5 # 5 # 5 # 5 and 1 -324 = 1 -32 # 1 -32 # 1 -32 # 1 -32

(+1+)++1* 6 factors; each factor is 5

(++++++)+1++++* 4 factors; each factor is -3

The base of an exponential expression is the repeated factor. The exponent is the

number of times that the base is used as a factor.

?

56

exponent base

1 -324

?

exponent base

? ?

E X A M P L E 1 Evaluate each expression. a. 23 b. 31 c. 1 - 422 d. - 42

14 e. ? 2

f. 10.523

g. 4 # 32

Solution

a. 23 = 2 # 2 # 2 = 8

b. To raise 3 to the first power means to use 3 as a factor only once. Therefore,

# 31 = 3. Also, when no exponent is shown, the exponent is assumed to be 1.

c. 1 - 422 = 1 - 42 1 - 42 = 16

d. - 42 = - 14 42 = - 16

e. g.

? 4

21# 324

= =

1

42#

#1#

2 9=

1#1

22 36

=

1 16

f. 10.523 = 10.52 10.52 10.52 = 0.125

PRACTICE

1 Evaluate each expression.

a. 33

b. 41

e.

?

3 4

3

f. 10.324

c. 1 - 822

g. 3 # 52

d. - 82

Notice how similar -42 is to 1 -422 in the example above. The difference between the two is the parentheses. In 1 -422, the parentheses tell us that the base, or repeated factor, is - 4. In - 42, only 4 is the base.

Helpful Hint

Be careful when identifying the base of an exponential expression. Pay close attention to the

use of parentheses. 1 -322

- 32

2 # 32

The base is - 3. 1 -322 = 1 -321 -32 = 9

-

32T=he-b1a3se#

is 3. 32 =

-9

2

#

The 32 =

b2a#s3e

#is33=.

18

An exponent has the same meaning whether the base is a number or a variable.

If x is a real number and n is a positive integer, then xn is the product of n factors, each

of which is x.

xn = x # x # x # x # x # c # x

(++++)++++* n factors of x

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312 CHAPTER 5 Exponents and Polynomials

E X A M P L E 2 Evaluate each expression for the given value of x.

a. 2x3; x is 5

b.

9 x2

;

x

is

-3

Solution

a.

If x is 5,

2x3

= = =

2 2 2

# # #

11552# 35

125

#

52

= 250

b.

If x is -3,

9 x2

=

9 1 -322

9 = 1 -321 -32

9

= 9

=1

PRACTICE

2 Evaluate each expression for the given value of x.

a. 3x4; x is 3

b. x62; x is - 4

OBJECTIVE

2 Using the Product Rule

Exponential expressions can be multiplied, divided, added, subtracted, and themselves

raised to powers. By our definition of an exponent,

54 # 53 = 15 # 5 # 5 # 52 # 15 # 5 # 52

(11+)1111* (111)11*

=

54#f5ac# t5o#rs5 o# 5f 5# 5

#

3 5

factors

of

5

(11+1+)++111* 7 factors of 5

= 57

Also,

x2

#

x3

= =

x1x#

x# x# 2x

# #

1xx#

#x

x

#

x2

= x5

In both cases, notice that the result is exactly the same if the exponents are added.

#54 53 = 54 + 3 = 57

and

#x2 x3 = x2 + 3 = x5

This suggests the following rule.

? ?

?

Product Rule for Exponents If m and n are positive integers and a is a real number, then

# am an = am + n d Add exponents.

Keep common base.

# For example, 35 37 = 35 + 7 = 312 d Add exponents.

Keep common base.

Helpful Hint Don't forget that

#35 37 912 d Add exponents. 35 # 37 = 3 # 3 # 3 C# 3o#m3m~ 3on# 3b#a3se# 3no# 3t k# e3p# t3.

(++)++* (++++)++++* 5 factors of 3 7 factors of 3

= 312 12 factors of 3, not 9

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Section 5.1 Exponents 313

In other words, to multiply two exponential expressions with the same base, we keep the base and add the exponents. We call this simplifying the exponential expression.

EXAMPLE 3

a. d.

42 y3

# #

45 y2

#

y7

Use the product rule to simplify.

b. e.

x4 # x6

1 -527

#

1 -528

c. f.

y3 a2

# #

y b2

Solution

# a. 42 45 = 42 + 5 = 47 d Add exponents.

?

# b. x4 x6 = x4 + 6 = x10 # # c. y3 y = y3 y1

= y3 + 1

Keep common base. Helpful Hint

Don't forget that if no exponent is written, it is assumed to be 1.

= y4

# # d. y3 y2 y7 = y3 + 2 + 7 = y12 # e. 1 - 527 1 - 528 = 1 - 527 + 8 = 1 - 5215 # f. a2 b2 Cannot be simplified because a and b are different bases.

PRACTICE

3

aecU... s3z1e4-# t#z2h342e65p#r1o-du22ct3 rule to simplify.

b. d. f.

y3 x3 b3

# # #

y2 x2 t5

#

x6

CONCEPT CHECK

# # # Where possible, use the product rule to simplify the expression.

a. z2 z14

b. x2 y14

c. 98 93

d. 98 # 27

E X A M P L E 4 Use the product rule to simplify 12x 22 1 -3x 52.

Solution Recall that 2x2 means 2 # x2 and -3x5 means -3 # x5. # # # 12x22 1 - 3x52 = 2 x2 - 3 x5 Remove parentheses. # # # = 2 - 3 x2 x5 Group factors with common bases.

= - 6x7

Simplify.

PRACTICE

4 Use the product rule to simplify 1 - 5y32 1 - 3y42.

Answers to Concept Check:

a. z16

b. cannot be simplified

c. 911

d. cannot be simplified

E X A M P L E 5 Simplify.

a. 1x2y2 1x3y22

b. 1 - a7b42 13ab92

Solution

# # # a. 1x2y2 1x3y22 = 1x2 x32 1y1 y22 Group like bases and write y as y1. # = x5 y3 or x5y3 Multiply.

b. 1 -a7b4213ab92 = 1 -1 # 32 # 1a7 # a12 # 1b4 # b92

= - 3a8b13

PRACTICE

5 Simplify. a. 1y7z32 1y5z2

b. 1 - m4n42 17mn102

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314 CHAPTER 5 Exponents and Polynomials

Helpful Hint

These examples will remind you of the difference between adding and multiplying terms. Addition

5x3 + 3x3 = 15 + 32x3 = 8x3 By the distributive property.

7x + 4x2 = 7x + 4x2

Cannot be combined.

Multiplication

# # # 15x32 13x32 = 5 3 x3 x3 = 15x3 + 3 = 15x6 17x214x22 = 7 # 4 # x # x2 = 28x1 + 2 = 28x3

By the product rule. By the product rule.

OBJECTIVE

3 Using the Power Rule

Exponential expressions can themselves be raised to powers. Let's try to discover a rule that simplifies an expression like 1x223. By definition,

1x223 = 1x221x221x22 (11+)1+1* 3 factors of x2

which can be simplified by the product rule for exponents. 1x223 = 1x221x221x22 = x2 + 2 + 2 = x6

Notice that the result is exactly the same if we multiply the exponents.

1x223 = x2 # 3 = x6

The following property states this result.

? ?

Power Rule for Exponents If m and n are positive integers and a is a real number, then

1am2n = amn d Multiply exponents. Keep common base.

For example, 17225 = 72 # 5 = 710 d Multiply exponents.

Keep common base.

To raise a power to a power, keep the base and multiply the exponents.

EXAMPLE 6 a. 1y822

Use the power rule to simplify. b. 18425

Solution a.

1y822

=

y8 # 2

=

y16

b. 18425 = 84#5 = 820

PRACTICE

6 Use the power rule to simplify.

a. 1z327

b. 14922

c. 3 1 - 52347 c. 3 1 - 52347 = 1 - 5221

c. 3 1 - 22345

Helpful Hint

Take a moment to make sure that you understand when to apply the product rule and when to apply the power rule.

Product Rule S Add Exponents

#x5 x7 = x5 + 7 = x12 #y6 y2 = y6 + 2 = y8

Power Rule S Multiply Exponents

1x527 = x5 # 7 = x35 1y622 = y6 # 2 = y12

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