3.1
Exponents and Polynomials
3.1
OBJECTIVES
1.
2.
3.
4.
5.
6.
Recognize the five properties of exponents
Use the properties to simplify expressions
Identify types of polynomials
Find the degree of a polynomial
Write a polynomial in descending exponent form
Evaluate a polynomial
Overcoming Math Anxiety
Hint #4
Preparing for a Test
Preparation for a test really begins on the first day of class. Everything you have
done in class and at home has been part of that preparation. However, there
are a few things that you should focus on in the last few days before a
scheduled test.
1. Plan your test preparation to end at least 24 hours before the test. The last
24 hours is too late, and besides, you will need some rest before the test.
2. Go over your homework and class notes with pencil and paper in hand. Write
down all of the problem types, formulas, and definitions that you think might
give you trouble on the test.
3. The day before the test, take the page(s) of notes from step 2, and transfer the
most important ideas to a 3 5 card.
4. Just before the test, review the information on the card. You will be surprised
at how much you remember about each concept.
5. Understand that, if you have been successful at completing your homework
assignments, you can be successful on the test. This is an obstacle for many
students, but it is an obstacle that can be overcome. Truly anxious students are
often surprised that they scored as well as they did on a test. They tend to
attribute this to blind luck. It is not. It is the first sign that you really do “get
it.” Enjoy the success.
In Chapter 0, we introduced the idea of exponents. Recall that the exponent notation indicates repeated multiplication and that the exponent tells us how many times the base is to
be used as a factor.
Exponent
35 3 3 3 3 3 243
5 factors
© 2001 McGraw-Hill Companies
Base
Now, we will look at the properties of exponents.
The first property is used when multiplying two values with the same base.
Rules and Properties: Property 1 of Exponents
For any real number a and positive integers m and n,
am an amn
For example,
25 27 212
247
248
CHAPTER 3
POLYNOMIALS
The second property is used when dividing two values with the same base.
Rules and Properties: Property 2 of Exponents
For any real number a and positive integers m and n, with m n,
aman amn
For example,
21227 25
Consider the following:
NOTE Notice that this means
that the base, x2, is used as a
factor 4 times.
(x2)4 x2 x2 x2 x2 x8
This leads us to our third property for exponents.
Rules and Properties: Property 3 of Exponents
For any real number a and positive integers m and n,
(am)n amn
For example,
(23)2 232 26
The use of this new property is illustrated in Example 1.
Example 1
Using the Third Property of Exponents
C A U TI ON
Be careful! Be sure to distinguish
between the correct use of
Property 1 and Property 3.
(x4)5 x 45 x 20
Simplify each expression.
(a) (x4)5 x45 x20
(b) (2 ) 2
3 4
34
Multiply the exponents.
2
12
x4 x5 x 45 x9
CHECK YOURSELF 1
Simplify each expression.
(a) (m5)6
(b) (m5)(m6)
(c) (32)4
(d) (32)(34)
Suppose we now have a product raised to a power. Consider an expression such as
NOTE Here the base is 3x.
(3x)4
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but
EXPONENTS AND POLYNOMIALS
SECTION 3.1
249
We know that
(3x)4 (3x)(3x)(3x)(3x)
NOTE Here we have applied
the commutative and
associative properties.
(3 3 3 3)(x x x x)
34 x4 81x4
Note that the power, here 4, has been applied to each factor, 3 and x. In general, we have
Rules and Properties: Property 4 of Exponents
For any real numbers a and b and positive integer m,
(ab)m ambm
For example,
(3x)3 33 x3 27x3
The use of this property is shown in Example 2.
Example 2
5
NOTE Notice that (2x) and 2x
are entirely different
expressions. For (2x)5, the base
is 2x, so we raise each factor to
the fifth power. For 2x5, the
base is x, and so the exponent
applies only to x.
5
Using the Fourth Property of Exponents
Simplify each expression.
(a) (2x)5 25 x5 32x5
(b) (3ab)4 34 a4 b4 81a4b4
(c) 5(2r)3 5 23 r3 40r3
CHECK YOURSELF 2
Simplify each expression.
(a) (3y)4
(b) (2mn)6
(c) 3(4x)2
(d) 5x3
We may have to use more than one of our properties in simplifying an expression
involving exponents. Consider Example 3.
Example 3
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NOTE To help you understand
each step of the simplification,
we refer to the property being
applied. Make a list of the
properties now to help you as
you work through the
remainder of this and the next
section.
Using the Properties of Exponents
Simplify each expression.
(a) (r4s3)3 (r4)3 (s3)3
r s
12 9
Property 4
Property 3
(b) (3x2)2 (2x3)3
32(x2)2 23 (x3)3
Property 4
9x4 8x9
Property 3
72x
Multiply the coefficients and apply Property 1.
13
CHAPTER 3
POLYNOMIALS
(c)
(a3)5
a15
4 a
a4
Property 3
a11
Property 2
CHECK YOURSELF 3
Simplify each expression.
(a) (m5n2)3
(b) (2p)4(4p2)2
(c)
(s4)3
s5
We have one final exponent property to develop. Suppose we have a quotient raised to a
power. Consider the following:
3
x
3
x x x
xxx
x3
3
3 3 3
333
3
Note that the power, here 3, has been applied to the numerator x and to the denominator 3.
This gives us our fifth property of exponents.
Rules and Properties: Property 5 of Exponents
For any real numbers a and b, when b is not equal to 0, and positive integer m,
b
a
m
am
bm
For example,
5
2
3
23
8
53
125
Example 4 illustrates the use of this property. Again note that the other properties may
also have to be applied in simplifying an expression.
Example 4
Using the Fifth Property of Exponents
Simplify each expression.
3
4
3
x3
4
(a)
(b)
y (c)
33
27
43
64
(x3)4
(y2)4
x12
8
y
2 3 2
rs
(r2s3)2
t4
(t4)2
2
Property 5
Property 5
Property 3
Property 5
(r2)2(s3)2
(t4)2
Property 4
r4s6
t8
Property 3
© 2001 McGraw-Hill Companies
250
EXPONENTS AND POLYNOMIALS
SECTION 3.1
251
CHECK YOURSELF 4
Simplify each expression.
(a)
2
3
4
(b)
m3
n4
5
(c)
a2b3
c5
2
The following table summarizes the five properties of exponents that were discussed in
this section:
General Form
Example
1. a a a
x2 x3 x5
am
amn (m n)
an
3. (am)n amn
57
54
53
(z5)4 z20
(4x)3 43x3 64x3
m n
mn
2.
4. (ab)m ambm
5.
a
b
m
am
bm
2
3
6
26
64
36
729
Our work in this chapter deals with the most common kind of algebraic expression, a
polynomial. To define a polynomial, let’s recall our earlier definition of the word term.
Definitions: Term
A term is a number or the product of a number and one or more variables.
For example, x5, 3x, 4xy2, and 8 are terms. A polynomial consists of one or more terms
in which the only allowable exponents are the whole numbers, 0, 1, 2, 3, . . . and so on.
These terms are connected by addition or subtraction signs.
Definitions: Numerical Coefficient
© 2001 McGraw-Hill Companies
NOTE In a polynomial, terms
are separated by and signs.
In each term of a polynomial, the number is called the numerical coefficient, or
more simply the coefficient, of that term.
Example 5
Identifying Polynomials
(a) x 3 is a polynomial. The terms are x and 3. The coefficients are 1 and 3.
(b) 3x2 2x 5, or 3x2 (2x) 5, is also a polynomial. Its terms are 3x2, 2x,
and 5. The coefficients are 3, 2, and 5.
3
(c) 5x3 2 is not a polynomial because of the division by x in the third term.
x
252
CHAPTER 3
POLYNOMIALS
CHECK YOURSELF 5
Which of the following are polynomials?
(b) 3y3 2y (a) 5x2
5
y
(c) 4x2 2x 3
Certain polynomials are given special names because of the number of terms that they
have.
Definitions: Monomial, Binomial, and Trinomial
NOTE The prefix mono- means
1. The prefix bi- means 2. The
prefix tri- means 3. There are no
special names for polynomials
with four or more terms.
A polynomial with one term is called a monomial.
A polynomial with two terms is called a binomial.
A polynomial with three terms is called a trinomial.
Example 6
Identifying Types of Polynomials
(a) 3x2y is a monomial. It has one term.
(b) 2x3 5x is a binomial. It has two terms, 2x3 and 5x.
(c) 5x2 4x 3, or 5x2 (4x) 3, is a trinomial. Its three terms are 5x2, 4x,
and 3.
CHECK YOURSELF 6
Classify each of these as a monomial, binomial, or trinomial.
(a) 5x4 2x3
NOTE Remember, in a
polynomial the allowable
exponents are the whole
numbers 0, 1, 2, 3, and so on.
The degree will be a whole
number.
(b) 4x7
(c) 2x2 5x 3
We also classify polynomials by their degree. The degree of a polynomial that has only
one variable is the highest power appearing in any one term.
Example 7
Classifying Polynomials by Their Degree
The highest power
The highest power
NOTE We will see in the next
section that x0 1.
(b) 4x 5x4 3x3 2 has degree 4.
(c) 8x has degree 1.
(Because 8x 8x1)
(d) 7 has degree 0.
Note: Polynomials can have more than one variable, such as 4x2y3 5xy2. The degree is
then the sum of the highest powers in any single term (here 2 3, or 5). In general, we will
be working with polynomials in a single variable, such as x.
© 2001 McGraw-Hill Companies
(a) 5x3 3x2 4x has degree 3.
EXPONENTS AND POLYNOMIALS
SECTION 3.1
253
CHECK YOURSELF 7
Find the degree of each polynomial.
(a) 6x5 3x3 2
(b) 5x
(c) 3x3 2x6 1
(d) 9
Working with polynomials is much easier if you get used to writing them in
descending-exponent form (sometimes called descending-power form). This simply
means that the term with the highest exponent is written first, then the term with the next
highest exponent, and so on.
Example 8
Writing Polynomials in Descending Order
The exponents get smaller
from left to right.
(a) 5x7 3x4 2x2 is in descending-exponent form.
(b) 4x4 5x6 3x5 is not in descending-exponent form. The polynomial should be
written as
5x6 3x5 4x4
Notice that the degree of the polynomial is the power of the first, or leading, term once the
polynomial is arranged in descending-exponent form.
CHECK YOURSELF 8
Write the following polynomials in descending-exponent form.
(a) 5x4 4x5 7
(b) 4x3 9x4 6x8
A polynomial can represent any number. Its value depends on the value given to the
variable.
Example 9
Evaluating Polynomials
© 2001 McGraw-Hill Companies
Given the polynomial
3x3 2x2 4x 1
(a) Find the value of the polynomial when x 2.
Substituting 2 for x, we have
NOTE Again note how the
rules for the order of
operations are applied. See
Section 0.3 for a review.
3(2)3 2(2)2 4(2) 1
3(8) 2(4) 4(2) 1
24 8 8 1
9
254
CHAPTER 3
POLYNOMIALS
(b) Find the value of the polynomial when x 2.
Be particularly careful when
dealing with powers of
negative numbers!
Now we substitute 2 for x.
3(2)3 2(2)2 4(2) 1
3(8) 2(4) 4(2) 1
24 8 8 1
23
CHECK YOURSELF 9
Find the value of the polynomial
4x3 3x2 2x 1
When
(a) x 3
(b) x 3
CHECK YOURSELF ANSWERS
1. (a) m30; (b) m11; (c) 38; (d) 36
3.
5.
7.
9.
2. (a) 81y4; (b) 64m6n6; (c) 48x2; (d) 5x3
16
m15
a4b6
(a) m15n6; (b) 256p8; (c) s7
4. (a) ; (b) 20 ; (c) 10
81
n
c
(a) and (c) are polynomials.
6. (a) Binomial; (b) monomial; (c) trinomial
(a) 5; (b) 1; (c) 6; (d) 0
8. (a) 4x5 5x4 7; (b) 6x8 9x4 4x3
(a) 86; (b) 142
© 2001 McGraw-Hill Companies
C A U TI ON
Name
Exercises
3.1
Section
Date
Use Property 3 of exponents to simplify each of the following expressions.
1. (x2)3
2. (a5)3
3. (m4)4
4. (p7)2
5. (24)2
6. (33)2
7. (53)5
8. (72)4
ANSWERS
Use the five properties of exponents to simplify each of the following expressions.
9. (3x)3
12. (5pq)3
15.
18.
3
4
14. 4(2rs)4
2
3
3
17.
x
5
19. (2x2)4
24. (4m n )
(x4)3
x2
5 3
20. (3y2)5
22. ( p3q4)2
3 2
25. (3m ) (m )
28.
3 2
(y ) (y )
(y4)4
23. (4x2y)3
2 4
(m5)3
m6
4 3
3 3
31.
3 2
26. (y ) (4y )
29.
(s3)2(s2)3
(s5)2
4 4
n m
32.
2
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
3
5
4 4 2
30.
13. 5(3ab)3
16.
21. (a8b6)2
27.
11. (2xy)4
2
a
2
10. (4m)2
1.
b a
35.
3
36.
3 2 2
33.
c ab
4
5 2 3
34.
z xy
37.
4
38.
© 2001 McGraw-Hill Companies
Which of the following expressions are polynomials?
35. 7x3
36. 5x3 38. 7
39. 7
3
x
39.
37. 4x4y2 3x3y
40.
41.
41.
3x
x2
40. 4x3 x
42.
42. 5a2 2a 7
255
ANSWERS
43.
For each of the following polynomials, list the terms and the coefficients.
44.
45.
46.
43. 2x2 3x
44. 5x3 x
45. 4x3 3x 2
46. 7x2
47.
Classify each of the following as a monomial, binomial, or trinomial where possible.
48.
49.
50.
47. 7x3 3x2
48. 4x7
49. 7y2 4y 5
50. 2x2 3xy y2
51. 2x4 3x2 5x 2
4
52. x 53. 6y8
54. 4x4 2x2 5x 7
51.
52.
53.
5
7
x
54.
55.
5
55. x 3
x2
56. 4x2 9
56.
Arrange in descending-exponent form if necessary, and give the degree of each
polynomial.
57.
58.
59.
60.
57. 4x5 3x2
58. 5x2 3x3 4
59. 7x7 5x9 4x3
60. 2 x
61. 4x
62. x17 3x4
63. 5x2 3x5 x6 7
64. 5
61.
62.
63.
64.
66.
67.
68.
69.
70.
71.
72.
256
65. 6x 1, x 1 and x 1
66. 5x 5, x 2 and x 2
67. x3 2x, x 2 and x 2
68. 3x2 7, x 3 and x 3
69. 3x2 4x 2, x 4 and x 4
70. 2x2 5x 1, x 2 and x 2
71. x2 2x 3, x 1 and x 3
72. x2 5x 6, x 3 and x 2
© 2001 McGraw-Hill Companies
Find the values of each of the following polynomials for the given values of the variable.
65.
ANSWERS
Indicate whether each of the following statements is always true, sometimes true, or never
true.
73. A monomial is a polynomial.
74. A binomial is a trinomial.
75. The degree of a trinomial is 3.
76. A trinomial has three terms.
77. A polynomial has four or more terms.
78. A binomial must have two
coefficients.
73.
74.
75.
76.
77.
78.
Solve the following problems.
79.
79. Write x12 as a power of x2.
80. Write y15 as a power of y3.
80.
81. Write a16 as a power of a2.
82. Write m20 as a power of m5.
81.
83. Write each of the following as a power of 8. (Remember that 8 23.)
12
18
5 3
82.
7 6
2 , 2 , (2 ) , (2 )
84. Write each of the following as a power of 9.
83.
84.
38, 314, (35)8, (34)7
85.
85. What expression raised to the third power is 8x6y9z15?
86.
86. What expression raised to the fourth power is 81x12y8z16?
The formula (1 R)Y G gives us useful information about the growth of a population.
Here R is the rate of growth expressed as a decimal, y is the time in years, and G is the
growth factor. If a country has a 2 percent growth rate for 35 years, then it will double its
population:
(1.02)35 2
87.
88.
89.
87. a. With this growth rate, how many doublings will occur in 105 years? How much
larger will the country’s population be?
90.
© 2001 McGraw-Hill Companies
b. The less developed countries of the world had an average growth rate of 2 percent
in 1986. If their total population was 3.8 billion, what will their population be in
105 years if this rate remains unchanged?
88. The United States has a growth rate of 0.7 percent. What will be its growth factor
after 35 years?
89. Write an explanation of why (x3)(x4) is not x12.
90. Your algebra study partners are confused. “Why isn’t x2 x3 2x5?”, they ask you.
Write an explanation that will convince them.
257
ANSWERS
91.
Capital italic letters such as P or Q are often used to name polynomials. For example, we
might write P(x) 3x3 5x2 2 in which P(x) is read “P of x.” The notation permits a
convenient shorthand. We write P(2), read “P of 2,” to indicate the value of the
polynomial when x 2. Here
92.
93.
P(2) 3(2)3 5(2)2 2
94.
38542
95.
6
96.
Use the information above in the following problems.
97.
If P(x) x3 2x2 5 and Q(x) 2x2 3, find:
98.
91. P(1)
92. P(1)
93. Q(2)
94. Q(2)
99.
100.
95. P(3)
101.
98. Q(0)
102.
96. Q(3)
99. P(2) Q(1)
97. P(0)
100. P(2) Q(3)
101. P(3) Q(3) Q(0) 102. Q(2) Q(2) P(0)
103.
103. Q(4) P(4)
104.
104.
P(1) Q(0)
P(0)
105.
105. Cost of typing. The cost, in dollars, of typing a term paper is given as 3 times the
number of pages plus 20. Use y as the number of pages to be typed and write a
polynomial to describe this cost. Find the cost of typing a 50-page paper.
106.
106. Manufacturing. The cost, in dollars, of making suits is described as 20 times the
© 2001 McGraw-Hill Companies
number of suits plus 150. Use s as the number of suits and write a polynomial to
describe this cost. Find the cost of making seven suits.
258
ANSWERS
107. Revenue. The revenue, in dollars, when x pairs of shoes are sold is given by 3x2 95.
Find the revenue when 12 pairs of shoes are sold. What is the average revenue per pair
of shoes?
108. Manufacturing. The cost in dollars of manufacturing w wing nuts is given by the
expression 0.07w 13.3. Find the cost when 375 wing nuts are made. What is the
average cost to manufacture one wing nut?
107.
108.
109.
110.
109. Suppose that when you were born, a rich uncle put $500 in the bank for you. He never
deposited money again, but the bank paid 5 percent interest on the money every year
on your birthday. How much money was in the bank after 1 year? After 2 years? After
1 year (as you know), the amount is $500 500(0.05), which can be written as
$500(1 0.05) because of the distributive property. 1 0.05 1.05, so after 1 year
the amount in the bank was 500(1.05). After 2 years, this amount was again multiplied
by 1.05. How much is in the bank today? Complete the following chart.
Birthday
Computation
0
(Day of Birth)
1
Amount
$500
$500(1.05)
2
$500(1.05)(1.05)
3
$500(1.05)(1.05)(1.05)
4
$500(1.05)4
5
$500(1.05)5
6
7
8
Write a formula for the amount in the bank on your nth birthday. About how many
years does it take for the money to double? How many years for it to double again? Can
you see any connection between this and the rules for exponents? Explain why you
think there may or may not be a connection.
110. Work with another student to correctly complete the statements:
© 2001 McGraw-Hill Companies
(a)
m3
1
n3
ax
1
ay
when . . .
m3
1 when . . .
n3
ax
1
ay
when . . .
m3
1 when . . .
n3
ax
1
ay
when . . .
m3
0
n3
ax
0
ay
when . . .
ax
0
ay
when . . .
when . . .
(is negative) when . . .
m3
0 when . . .
n3
(b)
259
ANSWERS
a.
Getting Ready for Section 3.2 [Section 1.7]
b.
Reduce each of the following fractions to simplest form.
c.
d.
(a)
m3
m5
(b)
x7
x10
(c)
a3
a9
(d)
y4
y8
(e)
x3
x3
(f)
b5
b5
(g)
s7
s7
(h)
r10
r10
e.
f.
Answers
g.
1. x6
h.
15.
3. m16
9
16
17.
91. 4
x
125
29. s2
93. 11
105. 3y 20, $170
9. 27x3
11. 16x4y4
13. 135a3b3
19. 16x8
21. a16b12
23. 64x6y3
25. 81m14
1
a6
d.
1
y4
31.
95. 14
97. 5
107. $337, $28.08
e. 1
f. 1
g. 1
99. 10
109.
103. 2
101. 7
a.
1
m2
b.
1
x3
h. 1
© 2001 McGraw-Hill Companies
c.
7. 515
m9
a6b4
33.
35. Polynomial
6
n
c8
Polynomial
39. Polynomial
41. Not a polynomial
2x2, 3x; 2, 3
45. 4x3, 3x, 2; 4, 3, 2
47. Binomial
Trinomial
51. Not classified
53. Monomial
55. Not a polynomial
4x5 3x2; 5
59. 5x9 7x7 4x3; 9
61. 4x; 1
x6 3x5 5x2 7; 6
65. 7, 5
67. 4, 4
69. 62, 30
0, 0
73. Always
75. Sometimes
77. Sometimes
79. (x2)6
2 8
4
6
5
14
2 3 5
(a )
83. 8 , 8 , 8 , 8
85. 2x y z
(a) Three doublings, 8 times as large; (b) 30.4 billion
89.
27. x10
37.
43.
49.
57.
63.
71.
81.
87.
5. 28
3
260