Chapter 5 Exponents and Polynomials

For use by Palm Beach State College only.

Chapter

5

Exponents and Polynomials

5.1 Exponents

5.2 Adding and Subtracting

?Polynomials

5.3 Multiplying Polynomials

5.4 Special Products

Integrated Review¡ªExponents

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

Can You Imagine a World Without the Internet?

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.

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

Worldwide Internet Users

(by region of the world)

Oceania

Europe

Asia

Africa

Americas

Number of Internet Users (in milions)

3500

3000

2500

2000

1500

1000

500

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

Objective

Objectives

1 Evaluate Exponential

?Expressions.

2 Use the Product Rule for

?Exponents.

Evaluating Exponential Expressions

1

As we reviewed in Section 1.4, an exponent is a shorthand notation for repeated factors.

For example, 2 # 2 # 2 # 2 # 2 can be written as 25. The expression 25 is called an exponential 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+)++1*

6 factors; each factor is 5

3 Use the Power Rule for

?Exponents.

4 Use the Power Rules for

?Products and Quotients.

? exponent

base

exponent

base

1 -32 4

?

6 Decide Which Rule(s) to Use to

56

?

?Exponents, and Define a

?Number Raised to the

0 Power.

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.

?

5 Use the Quotient Rule for

1 -32 4 = 1 -32 # 1 -32 # 1 -32 # 1 -32

(++++++)+1++++*

4 factors; each factor is -3

E x a m p l e 1 Evaluate each expression.

Simplify an Expression.

a. 23

b. 31

c. 1 -42 2

d. -42

1 4

e. ? ¡Ü

2

f. 10.52 3

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 -42 2 = 1 -421 -42 = 16

d. -42 = - 14 # 42 = -16

1 4

1 1 1 1

1

f. 10.52 3 = 10.5210.5210.52 = 0.125

e. ? ¡Ü = # # # =

2

2 2 2 2

16

g. 4 # 32 = 4 # 9 = 36 ?

Practice

1

Evaluate each expression.

a. 33

b. 41

3 3

e. ? ¡Ü

4

c. 1 -82 2

g. 3 # 52

f. 10.32 4

d. -82

?

Notice how similar -42 is to 1 -42 2 in the example above. The difference between

the two is the parentheses. In 1 -42 2, 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 -32 2

The base is - 3.

1 - 32 2 = 1 - 321 -32 = 9

- 32

The base is 3.

- 32 = - 13 # 32 = - 9

2 # 32

The base is 3.

2 # 32 = 2 # 3 # 3 = 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 x n is the product of n factors, each

of which is x.

#x

x n = (++++)++

x#x#x#x#x# c

++*

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. 2x 3; x is 5

b.

Solution??a. If x is 5, 2x 3 =

=

=

=

9

x2

; x is -3

2 # 152 3

2 # 15 # 5 # 52

2 # 125

250

b. If x is -3,

9

x

2

9

=

1 -32 2

9

=

1 -321 -32

9

=

9

= 1

?

Practice

2

Evaluate each expression for the given value of x.

6

a. 3x 4; x is 3

b. 2 ; x is -4

x

?

Objective

Using the Product Rule

2

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)111*

4 factors of 5 3 factors of 5

= 5#5#5#5#5#5#5

(11+1+)++111*

7 factors of 5

7

= 5

Also,

x 2 # x 3 = 1x # x2 # 1x # x # x2

= x#x#x#x#x

= 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

?

a m # a n = a m + n d Add exponents.

Keep common base.

?

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

Keep common base.

Helpful Hint

35 # 37 ¡Ù 912 d Add exponents.

Common base not kept.

?

Don¡¯t forget that

# +)++

# 3 # 3+

# 3*

35 # 37 = 3(+

3 # 3 # 3 # 3* ~ (+

3 #++

3 # +)

3 # 3 +++

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

Use the product rule to simplify.

a. 4 # 4

b. x 4 # x 6

c. y3 # y

d. y3 # y2 # y7

e. 1 -52 7 # 1 -52 8

f. a 2 # b2

Solution??

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

Keep common base.

4# 6

4+6

10

Helpful Hint

b. x x = x

= x

Don¡¯t forget that if no exponent

c. y3 # y = y3 # y1

is written, it is assumed to be 1.

3+1

= y

= y4

d. y3 # y2 # y7 = y3 + 2 + 7 = y12

e. 1 -52 7 # 1 -52 8 = 1 -52 7 + 8 = 1 -52 15

f. a 2 # b2 Cannot be simplified because a and b are different bases.?

5

?

2

Practice

3

Use the product rule to simplify.

a. 34 # 36

c. z # z4

e. 1 -22 5 # 1 -22 3

b. y3 # y2

d. x 3 # x 2 # x 6

f. b3 # t 5

?

Concept Check

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

a. z2 # z14

b. x 2 # y14

c. 98 # 93

Example 4

d. 98 # 27

Use the product rule to simplify 12x 2 21 -3x 5 2.

Solution Recall that 2x 2 means 2 # x 2 and -3x 5 means -3 # x 5.

12x 2 21 -3x 5 2 = 2 # x 2 # -3 # x 5

= 2 # -3 # x 2 # x 5

= -6x 7

Practice

4

Remove parentheses.

Group factors with common bases.

Simplify. ?

Example 5

Simplify.

a. 1x 2y21x 3y2 2

b. 1 -a 7b4 213ab9 2

Solution??

a. 1x 2y21x 3y2 2 = 1x 2 # x 3 2 # 1y1 # y2 2

= x 5 # y3 or x 5y3

Group like bases and write y as y1.

Multiply.

b. 1 -a 7b4 213ab9 2 = 1 -1 # 32 # 1a 7 # a 1 2 # 1b4 # b9 2

= -3a 8b13

Answers to Concept Check:

a. z16

b. cannot be simplified

c. 911

d. cannot be simplified

?

Practice

5

Simplify.

a. 1y7z3 21y5z2

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Use the product rule to simplify 1 -5y3 21 -3y4 2.

b. 1 -m4n4 217mn10 2

?

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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

5x 3 + 3x 3 = 15 + 32x 3 = 8x 3

2

7x + 4x = 7x + 4x

By the distributive property.

2

Cannot be combined.

Multiplication

15x 3 213x 3 2 = 5 # 3 # x 3 # x 3 = 15x 3 + 3 = 15x 6

2

17x214x 2 =

7 # 4 # x # x2

= 28x

1+2

= 28x

3

By the product rule.

By the product rule.

Objective

Using the Power Rule

3

Exponential expressions can themselves be raised to powers. Let¡¯s try to discover a

3

rule that simplifies an expression like 1x 2 2 . By definition,

1x 2 2 3 = 1x 2 21x 2 21x 2 2

(11+)1+1*

3 factors of x 2

which can be simplified by the product rule for exponents.

3

1x 2 2 = 1x 2 21x 2 21x 2 2 = x 2 + 2 + 2 = x 6

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

3

1x 2 2 = x 2

#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

n

?

1a m 2 = a mn d Multiply exponents.

#5

= 710 d Multiply exponents.

?

5

For example, 172 2 = 72

Keep common base.

Keep common base.

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

Example 6

Use the power rule to simplify.

8 2

a. 1y 2

Solution??

#

2

a. 1y8 2 = y8 2 = y16

Practice

6

b. 184 2 5

#

b. 184 2 5 = 8 4 5 = 820

Use the power rule to simplify.

a. 1z3 2 7

b. 149 2 2

Helpful Hint

c. 3 1 -52 3 4

7

7

c. 3 1 -52 3 4 = 1 -52 21 ?

c. 3 1 -22 3 4 5

?

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

#x = x

y6 # y2 = y6 + 2

x

5

7

5+7

= x

12

= y8

Power Rule S Multiply Exponents

7

#7

2

#2

1x 5 2 = x 5

1y6 2 = y6

= x 35

= y12

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