POLYNOMIALS
3
INTRODUCTION
The U.S. Post Office limits the size of rectangular boxes it will accept for mailing. The regulations state
that “length plus girth cannot exceed 108 inches.” “Girth”
means the distance around a cross section; in this case, this
measurement is 2h 2w. Using the polynomial l 2w 2h
to describe the measurement required by the Post Office, the
regulations say that l 2w 2h 108 inches.
The volume of a rectangular box is expressed by another
polynomial: V lwh
l
h
© 2001 McGraw-Hill Companies
w
A company that wishes to produce boxes for use by postal
patrons must use these formulas and do a statistical survey
about the shapes that are useful to the most customers. The surface area, expressed
by another polynomial expression, 2lw 2wh 2lh, is also used so each box can be
manufactured with the least amount of material, to help lower costs.
245
Name
Section
Pre-Test Chapter 3
Date
ANSWERS
Simplify each of the following expressions. Write your answers with positive exponents
only.
1.
2.
1. x5x7
3.
4.
2. (2x3y2) (4x2 y5)
3.
9x5y2
3x2y
4. (2x3y4)2 (x4 y6)0
5.
(x2)4
x8
6. 2x5y3
5.
6.
Classify each of the following polynomials as a monomial, binomial, or trinomial.
7.
7. 6x2 7x
8.
8. 4x3 5x 9
Add.
9.
9. 4x2 7x 5 and 2x2 5x 7
10.
11.
Subtract.
12.
10. 2x 2 3x 1 from 7x2 8x 5
13.
Multiply.
14.
11. 3xy(4x2y2 2xy 7xy3)
12. (3x 2)(2x 5)
13. (x 2y)(x 2y)
14. (4m 5)2
17.
15. (3x 2y)(x2 4xy 3y2)
16. x(3x 5y)2
18.
Divide.
15.
19.
17.
28x2y3 35x4y5
7x2y2
18.
x2 x 6
x3
19.
x3 2x 3x2 6
x2 2
20.
3x2 7x 25
x4
20.
246
© 2001 McGraw-Hill Companies
16.
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
CA UTIO N
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
4. (ab)m ambm
57
54
53
(z5)4 z20
(4x)3 43x3 64x3
m n
mn
2.
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
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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
CA UTI O N
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
3.2
Negative Exponents and
Scientific Notation
3.2
OBJECTIVES
1. Evaluate expressions involving zero or a negative
exponent
2. Simplify expressions involving zero or a negative
exponent
3. Write a decimal number in scientific notation
4. Solve an application of scientific notation
In Section 3.1, we discussed exponents.
We now want to extend our exponent notation to include 0 and negative integers as
exponents.
First, what do we do with x0? It will help to look at a problem that gives us x0 as a result.
What if the numerator and denominator of a fraction have the same base raised to the same
power and we extend our division rule? For example,
a5
a55 a0
a5
NOTE By Property 2,
am
amn
an
when m n. Here m and n are
both 5 so m n.
(1)
But from our experience with fractions we know that
a5
1
a5
(2)
By comparing equations (1) and (2), it seems reasonable to make the following definition:
Definitions: Zero Power
0
NOTE As was the case with ,
0
00 will be discussed in a
later course.
For any number a, a 0,
a0 1
In words, any expression, except 0, raised to the 0 power is 1.
Example 1 illustrates the use of this definition.
Example 1
Raising Expressions to the Zero Power
© 2001 McGraw-Hill Companies
CA UTI O N
In part (d ) the 0 exponent
applies only to the x and not to
the factor 6, because the base
is x.
Evaluate. Assume all variables are nonzero.
(a) 50 1
(b) 270 1
(c) (x2y)0 1
if x 0 and y 0
(d) 6x0 6 1 6
if x 0
CHECK YOURSELF 1
Evaluate. Assume all variables are nonzero.
(a) 70
(b) (8)0
(c) (xy3)0
(d) 3x0
261
262
CHAPTER 3
POLYNOMIALS
The second property of exponents allows us to define a negative exponent. Suppose that
the exponent in the denominator is greater than the exponent in the numerator. Consider
x2
the expression 5 .
x
Our previous work with fractions tells us that
NOTE Divide the numerator
and denominator by the two
common factors of x.
x2
xx
1
3
5 x
xxxxx
x
(1)
However, if we extend the second property to let n be greater than m, we have
REMEMBER:
am
amn
an
x2
x25 x3
x5
(2)
1
Now, by comparing equations (1) and (2), it seems reasonable to define x3 as 3 .
x
In general, we have this result:
Definitions: Negative Powers
For any number a, a 0, and any positive integer n,
NOTE John Wallis (1616–1703),
an English mathematician, was
the first to fully discuss the
meaning of 0 and negative
exponents.
an 1
an
Example 2
Rewriting Expressions That Contain Negative Exponents
Rewrite each expression, using only positive exponents.
Negative exponent in numerator
(a) x4 1
x4
Positive exponent
in denominator
(b) m7 1
m7
(c) 32 1
1
or
32
9
CA UTIO N
1
1
3 or
10
1000
(e) 2x3 2 1
2
3
x3
x
The 3 exponent applies only
to x, because x is the base.
(f)
a5
1
59
a4 4
9 a
a
a
(g) 4x5 4 1
4
5
x5
x
© 2001 McGraw-Hill Companies
(d) 103 NEGATIVE EXPONENTS AND SCIENTIFIC NOTATION
SECTION 3.2
263
CHECK YOURSELF 2
Write, using only positive exponents.
(a) a10
(b) 43
(c) 3x2
(d)
x5
x8
We will now allow negative integers as exponents in our first property for exponents.
Consider Example 3.
Example 3
Simplifying Expressions Containing Exponents
NOTE am an amn for any
Simplify (write an equivalent expression that uses only positive exponents).
integers m and n. So add the
exponents.
(a) x5x2 x5(2) x3
Note: An alternative approach would be
NOTE By definition
x2 1
x2
x5x2 x5 1
x5
3
2 2 x
x
x
(b) a7a5 a7(5) a2
(c) y5y9 y5 (9) y4 1
y4
CHECK YOURSELF 3
Simplify (write an equivalent expression that uses only positive exponents).
(a) x7x2
(b) b3b8
Example 4 shows that all the properties of exponents introduced in the last section can
be extended to expressions with negative exponents.
Example 4
Simplifying Expressions Containing Exponents
© 2001 McGraw-Hill Companies
Simplify each expression.
(a)
m3
m34
m4
m7 (b)
Property 2
1
m7
a2b6
a25b6(4)
a5b4
a7b10 b10
a7
Apply Property 2 to each variable.
264
CHAPTER 3
POLYNOMIALS
NOTE This could also be done
1
(2x4)3
Definition of the negative exponent
3
1
2 (x4)3
Property 4
1
8x12
Property 3
(c) (2x4)3 by using Property 4 first, so
(2x4)3 23 (x4)3 23x12
1
3 12
2x
1
8x12
(d)
(y2)4
y8
(y3)2
y6
Property 3
y8(6)
y2 Property 2
1
y2
CHECK YOURSELF 4
Simplify each expression.
(a)
x5
x3
(b)
m3n5
m2n3
(c) (3a3)4
(d)
(r3)2
(r4)2
Let us now take a look at an important use of exponents, scientific notation.
We begin the discussion with a calculator exercise. On most calculators, if you multiply
2.3 times 1000, the display will read
2300
Multiply by 1000 a second time. Now you will see
2300000.
Multiplying by 1000 a third time will result in the display
NOTE This must equal
2,300,000,000.
2.3
09
or
2.3
E09
And multiplying by 1000 again yields
NOTE Consider the following
2.3
12
or
2.3
E12
2.3 2.3 100
23 2.3 101
230 2.3 102
2300 2.3 103
23,000 2.3 104
230,000 2.3 105
Can you see what is happening? This is the way calculators display very large numbers.
The number on the left is always between 1 and 10, and the number on the right indicates
the number of places the decimal point must be moved to the right to put the answer in standard (or decimal) form.
This notation is used frequently in science. It is not uncommon in scientific applications
of algebra to find yourself working with very large or very small numbers. Even in the time
of Archimedes (287–212 B.C.E.), the study of such numbers was not unusual. Archimedes
estimated that the universe was 23,000,000,000,000,000 m in diameter, which is the
1
approximate distance light travels in 2 years. By comparison, Polaris (the North Star) is
2
actually 680 light-years from the earth. Example 6 will discuss the idea of light-years.
© 2001 McGraw-Hill Companies
table:
NEGATIVE EXPONENTS AND SCIENTIFIC NOTATION
SECTION 3.2
In scientific notation, Archimedes’s
estimate for the diameter of the universe
would be
2.3 1016 m
In general, we can define scientific notation as follows.
Definitions: Scientific Notation
Any number written in the form
a 10n
in which 1 a 10 and n is an integer, is written in scientific notation.
Example 5
Using Scientific Notation
Write each of the following numbers in scientific notation.
NOTE Notice the pattern for
writing a number in scientific
notation.
(a) 120,000. 1.2 105
5 places
The power is 5.
(b) 88,000,000. 8.8 107
The power is 7.
7 places
NOTE The exponent on 10
shows the number of places we
must move the decimal point.
A positive exponent tells us to
move right, and a negative
exponent indicates to move
left.
(c) 520,000,000. 5.2 108
8 places
(d) 4,000,000,000. 4 109
9 places
(e) 0.0005 5 104
If the decimal point is to be moved to
the left, the exponent will be negative.
© 2001 McGraw-Hill Companies
4 places
NOTE To convert back to
standard or decimal form, the
process is simply reversed.
(f) 0.0000000081 8.1 109
9 places
CHECK YOURSELF 5
Write in scientific notation.
(a) 212,000,000,000,000,000
(c) 5,600,000
(b) 0.00079
(d) 0.0000007
265
266
CHAPTER 3
POLYNOMIALS
Example 6
An Application of Scientific Notation
(a) Light travels at a speed of 3.05 108 meters per second (m/s). There are
approximately 3.15 107 s in a year. How far does light travel in a year?
We multiply the distance traveled in 1 s by the number of seconds in a year. This
yields
(3.05 108)(3.15 107) (3.05 3.15)(108 107)
9.6075 10
15
NOTE Notice that
9.6075 1015 10 1015 1016
Multiply the coefficients,
and add the exponents.
For our purposes we round the distance light travels in 1 year to 1016 m. This unit is
called a light-year, and it is used to measure astronomical distances.
(b) The distance from earth to the star Spica (in Virgo) is 2.2 1018 m. How many lightyears is Spica from earth?
Spica
2.2 × 1018 m
Earth
(in meters) by the number of
meters in 1 light-year.
2.2 1018
2.2 101816
1016
2.2 102 220 light-years
CHECK YOURSELF 6
The farthest object that can be seen with the unaided eye is the Andromeda galaxy.
This galaxy is 2.3 1022 m from earth. What is this distance in light-years?
CHECK YOURSELF ANSWERS
1
1
1
3
1
; (c) 2 ; (d) 3
10 ; (b) 3 or
a
4
64
x
x
5
1
m
1
3. (a) x5; (b) 5
4. (a) x8; (b) 8 ; (c)
; (d) r2
b
n
81a12
5. (a) 2.12 1017; (b) 7.9 104; (c) 5.6 106; (d) 7 107
6. 2,300,000 light-years
1. (a) 1; (b) 1; (c) 1; (d) 3
2. (a)
© 2001 McGraw-Hill Companies
NOTE We divide the distance
Name
3.2
Exercises
Section
Date
Evaluate (assume the variables are nonzero).
ANSWERS
0
1. 4
0
2. (7)
0
3. (29)
4. 750
5. (x3y2)0
6. 7m0
7. 11x0
8. (2a3b7)0
9. (3p6q8)0
10. 7x0
Write each of the following expressions using positive exponents; simplify when
possible.
11. b8
12. p12
13. 34
14. 25
15. 52
16. 43
17. 104
18. 105
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
19. 5x1
20. 3a2
21. (5x)1
22. (3a)2
23.
24.
23. 2x5
24. 3x4
25. (2x)5
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
26. (3x)4
Use Properties 1 and 2 to simplify each of the following expressions. Write your answers
with positive exponents only.
© 2001 McGraw-Hill Companies
1.
27. a5a3
28. m5m7
29. x8x2
30. a12a8
31. b7b11
32. y5y12
33. x0x5
34. r3r0
35.
35.
a8
a5
267
ANSWERS
36.
36.
m9
m4
37.
x7
x9
38.
a3
a10
39.
r3
r5
40.
x3
x5
41.
x4
x5
42.
p6
p3
37.
38.
39.
40.
41.
Simplify each of the following expressions. Write your answers with positive exponents
only.
42.
43.
43.
m5n3
m4n5
46. (3x2)3
44.
44.
p3q2
p4q3
47. (x2y3)2
45. (2a3)4
48. (a5b3)3
49.
(r2)3
r4
50.
(y3)4
y6
51.
(x3)3
(x4)2
52.
(m4)3
(m2)4
53.
(a3)2(a4)
(a3)3
54.
(x2)3(x2)
(x2)4
45.
46.
47.
48.
In exercises 55 to 58, express each number in scientific notation.
49.
55. The distance from the earth to the sun: 93,000,000 mi.
50.
51.
52.
53.
55.
56. The diameter of a grain of sand: 0.000021 m.
56.
57. The diameter of the sun: 130,000,000,000 cm.
57.
58. The number of molecules in 22.4 L of a gas: 602,000,000,000,000,000,000,000
58.
(Avogadro’s number).
59.
59. The mass of the sun is approximately 1.98 1030 kg. If this were written in standard
or decimal form, how many 0s would follow the digit 8?
268
© 2001 McGraw-Hill Companies
54.
ANSWERS
60. Archimedes estimated the universe to be 2.3 1019 millimeters (mm) in diameter. If
this number were written in standard or decimal form, how many 0s would follow the
digit 3?
60.
61.
62.
In exercises 61 to 64, write each expression in standard notation.
61. 8 103
62. 7.5 106
63. 2.8 105
64. 5.21 104
63.
64.
In exercises 65 to 68, write each of the following in scientific notation.
65. 0.0005
66. 0.000003
67. 0.00037
68. 0.000051
65.
66.
In exercises 69 to 72, compute the expressions using scientific notation, and write your
answer in that form.
67.
69. (4 103)(2 105)
68.
71.
70. (1.5 106)(4 102)
9 103
3 102
72.
7.5 104
1.5 102
In exercises 73 to 78, perform the indicated calculations. Write your result in scientific
notation.
73. (2 105)(4 104)
76.
4.5 1012
1.5 107
74. (2.5 107)(3 105)
77.
75.
6 109
3 107
(3.3 1015)(6 1015)
(6 1012)(3.2 108)
78.
8
6
(1.1 10 )(3 10 )
(1.6 107)(3 102)
69.
70.
71.
72.
73.
74.
75.
In 1975 the population of Earth was approximately 4 billion and doubling every 35 years.
The formula for the population P in year Y for this doubling rate is
P (in billions) 4 2(Y1975)35
76.
77.
78.
79. What was the approximate population of Earth in 1960?
80. What will Earth’s population be in 2025?
© 2001 McGraw-Hill Companies
The United States population in 1990 was approximately 250 million, and the average
growth rate for the past 30 years gives a doubling time of 66 years. The above formula for
the United States then becomes
79.
80.
81.
82.
P (in millions) 250 2(Y1990)66
81. What was the approximate population of the United States in 1960?
82. What will be the population of the United States in 2025 if this growth rate
continues?
269
ANSWERS
83.
83. Megrez, the nearest of the Big Dipper stars, is 6.6 1017 m from Earth.
Approximately how long does it take light, traveling at 1016 m/year, to travel from
Megrez to Earth?
84.
85.
84. Alkaid, the most distant star in the Big Dipper, is 2.1 1018 m from Earth.
Approximately how long does it take light to travel from Alkaid to Earth?
86.
87.
85. The number of liters (L) of water on Earth is 15,500 followed by 19 zeros. Write this
number in scientific notation. Then use the number of liters of water on Earth to find
out how much water is available for each person on Earth. The population of Earth is
6 billion.
a.
b.
86. If there are 6 109 people on Earth and there is enough freshwater to provide each
c.
person with 8.79 105 L, how much freshwater is on Earth?
d.
87. The United States uses an average of 2.6 106 L of water per person each year. The
e.
United States has 3.2 108 people. How many liters of water does the United States
use each year?
f.
g.
Getting Ready for Section 3.3 [Section 1.6]
h.
Combine like terms where possible.
(a)
(c)
(e)
(g)
8m 7m
9m2 8m
5c3 15c3
8c2 6c 2c2
(b) 9x 5x
(d) 8x2 7x2
(f) 9s3 8s3
(h) 8r3 7r2 5r3
Answers
3. 1
270
17.
1
10,000
7. 11
19.
5
x
9. 1
11.
1
b8
13.
1
81
1
2
23. 5
5x
x
1
1
1
29. x6
31. 4
33. x5
35. a3
37. 2
25. 27. a8
5
32x
b
x
1
m9
16
x4
1
1
43. 8
45. 12
47. 6
49. 2
51.
39. 8
41. x
r
n
a
y
r
x
1
7
11
57. 1.3 10 cm
59. 28
61. 0.008
53. 11
55. 9.3 10 mi
a
63. 0.000028
65. 5 104
67. 3.7 104
69. 8 108
5
9
2
71. 3 10
73. 8 10
75. 2 10
77. 6 1016
79. 2.97 billion
81. 182 million
83. 66 years
85. 1.55 1023 L; 2.58 1013 L
87. 8.32 1014 L
a. 15m
b. 4x
c. 9m2 8m
d. x2
e. 20c3
f. 17s3
g. 10c2 6c
h. 13r3 7r2
15.
1
25
5. 1
21.
© 2001 McGraw-Hill Companies
1. 1
3.3
Adding and Subtracting
Polynomials
3.3
OBJECTIVES
1. Add two polynomials
2. Subtract two polynomials
Addition is always a matter of combining like quantities (two apples plus three apples, four
books plus five books, and so on). If you keep that basic idea in mind, adding polynomials
will be easy. It is just a matter of combining like terms. Suppose that you want to add
5x 2 3x 4
and
4x 2 5x 6
Parentheses are sometimes used in adding, so for the sum of these polynomials, we can
write
NOTE The plus sign between
the parentheses indicates the
addition.
(5x 2 3x 4) (4x 2 5x 6)
Now what about the parentheses? You can use the following rule.
Rules and Properties: Removing Signs of Grouping Case 1
If a plus sign () or nothing at all appears in front of parentheses, just remove
the parentheses. No other changes are necessary.
Now let’s return to the addition.
NOTE Just remove the
parentheses. No other changes
are necessary.
(5x 2 3x 4) (4x 2 5x 6)
5x 2 3x 4 4x 2 5x 6
Like terms
NOTE Note the use of the
associative and commutative
properties in reordering and
regrouping.
Like terms
Like terms
Collect like terms. (Remember: Like terms have the same variables raised to the same
power).
(5x 2 4x2) (3x 5x) (4 6)
Combine like terms for the result:
NOTE Here we use the
distributive property. For
example,
5x 2 4x 2 (5 4)x 2
9x2 8x 2
As should be clear, much of this work can be done mentally. You can then write the sum
directly by locating like terms and combining. Example 1 illustrates this property.
© 2001 McGraw-Hill Companies
9x 2
Example 1
NOTE We call this the
“horizontal method” because
the entire problem is written on
one line.
3 4 7 is the horizontal
method.
3
4
7
is the vertical method.
Combining Like Terms
Add 3x 5 and 2x 3.
Write the sum.
(3x 5) (2x 3)
3x 5 2x 3 5x 2
Like terms
Like terms
271
272
CHAPTER 3
POLYNOMIALS
CHECK YOURSELF 1
Add 6x 2 2x and 4x 2 7x.
The same technique is used to find the sum of two trinomials.
Example 2
Adding Polynomials Using the Horizontal Method
Add 4a2 7a 5 and 3a2 3a 4.
Write the sum.
(4a2 7a 5) (3a2 3a 4)
REMEMBER Only the like
4a2 7a 5 3a2 3a 4 7a2 4a 1
terms are combined in the sum.
Like terms
Like terms
Like terms
CHECK YOURSELF 2
Add 5y 2 3y 7 and 3y 2 5y 7.
Example 3
Adding Polynomials Using the Horizontal Method
Add 2x 2 7x and 4x 6.
Write the sum.
(2x 2 7x) (4x 6)
2x 2 7x 4x 6
These are the only like terms;
2x 2 and 6 cannot be combined.
2x 2 11x 6
CHECK YOURSELF 3
As we mentioned in Section 3.1 writing polynomials in descending-exponent form
usually makes the work easier. Look at Example 4.
Example 4
Adding Polynomials Using the Horizontal Method
Add 3x 2x 2 7 and 5 4x 2 3x.
© 2001 McGraw-Hill Companies
Add 5m 2 8 and 8m 2 3m.
ADDING AND SUBTRACTING POLYNOMIALS
SECTION 3.3
273
Write the polynomials in descending-exponent form, then add.
(2x 2 3x 7) (4x 2 3x 5)
2x 2 12
CHECK YOURSELF 4
Add 8 5x 2 4x and 7x 8 8x 2.
Subtracting polynomials requires another rule for removing signs of grouping.
Rules and Properties: Removing Signs of Grouping Case 2
If a minus sign () appears in front of a set of parentheses, the parentheses can
be removed by changing the sign of each term inside the parentheses.
The use of this rule is illustrated in Example 5.
Example 5
Removing Parentheses
In each of the following, remove the parentheses.
(a) (2x 3y) 2x 3y
(2x 3y) (1)(2x 3y)
2x 3y
(b) m (5n 3p) m 5n 3p
Sign changes.
(c) 2x (3y z) 2x 3y z
NOTE This uses the distributive
property, because
Change each sign to remove the parentheses.
Sign changes.
CHECK YOURSELF 5
(a) (3m 5n)
(b) (5w 7z)
(c) 3r (2s 5t)
(d) 5a (3b 2c)
Subtracting polynomials is now a matter of using the previous rule to remove the
parentheses and then combining the like terms. Consider Example 6.
Subtracting Polynomials Using the Horizontal Method
(a) Subtract 5x 3 from 8x 2.
Write
NOTE The expression
following “from” is written first
in the problem.
(8x 2) (5x 3)
8x 2 5x 3
© 2001 McGraw-Hill Companies
Example 6
Sign changes.
3x 5
Recall that subtracting 5x is the same as adding 5x.
POLYNOMIALS
(b) Subtract 4x2 8x 3 from 8x2 5x 3.
Write
(8x 2 5x 3) (4x 2 8x 3)
8x 2 5x 3 4x 2 8x 3
Sign changes.
4x2 13x 6
CHECK YOURSELF 6
(a) Subtract 7x 3 from 10x 7.
(b) Subtract 5x 2 3x 2 from 8x 2 3x 6.
Again, writing all polynomials in descending-exponent form will make locating and
combining like terms much easier. Look at Example 7.
Example 7
Subtracting Polynomials Using the Horizontal Method
(a) Subtract 4x 2 3x 3 5x from 8x 3 7x 2x 2.
Write
(8x3 2x2 7x) (3x3 4x2 5x)
= 8x3 2x2 7x 3x3 4x2 5x
Sign changes.
11x3 2x2 12x
(b) Subtract 8x 5 from 5x 3x 2.
Write
(3x 2 5x) (8x 5)
3x 2 5x 8x 5
Only the like terms can be combined.
3x2 13x 5
CHECK YOURSELF 7
(a) Subtract 7x 3x 2 5 from 5 3x 4x 2.
(b) Subtract 3a 2 from 5a 4a2.
If you think back to addition and subtraction in arithmetic, you’ll remember that the
work was arranged vertically. That is, the numbers being added or subtracted were placed
under one another so that each column represented the same place value. This meant that in
adding or subtracting columns you were always dealing with “like quantities.”
© 2001 McGraw-Hill Companies
CHAPTER 3
274
ADDING AND SUBTRACTING POLYNOMIALS
SECTION 3.3
275
It is also possible to use a vertical method for adding or subtracting polynomials. First
rewrite the polynomials in descending-exponent form, then arrange them one under another, so that each column contains like terms. Then add or subtract in each column.
Example 8
Adding Using the Vertical Method
Add 2x2 5x, 3x2 2, and 6x 3.
Like terms
2x2 5x
3x2
2
6x 3
5x2 x 1
CHECK YOURSELF 8
Add 3x 2 5, x 2 4x, and 6x 7.
The following example illustrates subtraction by the vertical method.
Example 9
Subtracting Using the Vertical Method
(a) Subtract 5x 3 from 8x 7.
Write
8x 7
() 5x 3
3x 4
To subtract, change each
sign of 5x 3 to get
5x 3, then add.
8x 7
5x 3
3x 4
(b) Subtract 5x 2 3x 4 from 8x2 5x 3.
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Write
8x 2 5x 3
() 5x2 3x 4
3x2 8x 7
To subtract, change each
sign of 5x2 3x 4 to get
5x2 3x 4, then add.
8x 2 5x 3
5x2 3x 4
3x2 8x 7
Subtracting using the vertical method takes some practice. Take time to study the
method carefully. You’ll be using it in long division in Section. 3.6.
CHAPTER 3
POLYNOMIALS
CHECK YOURSELF 9
Subtract, using the vertical method.
(a) 4x 2 3x from 8x 2 2x
(b) 8x 2 4x 3 from 9x 2 5x 7
CHECK YOURSELF ANSWERS
1.
5.
6.
8.
10x2 5x
2. 8y2 8y
3. 13m2 3m 8
3. 3x2 11x
(a) 3m 5n; (b) 5w 7z; (c) 3r 2s 5t; (d) 5a 3b 2c
(a) 3x 10; (b) 3x 2 8
7. (a) 7x 2 10x; (b) 4a 2 2a 2
4x 2 2x 12
9. (a) 4x 2 5x; (b) x 2 9x 10
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276
Name
3.3
Exercises
Section
Date
Add.
ANSWERS
1. 6a 5 and 3a 9
2. 9x 3 and 3x 4
1.
3. 8b2 11b and 5b2 7b
4. 2m2 3m and 6m2 8m
2.
3.
5. 3x 2 2x and 5x 2 2x
6. 3p2 5p and 7p2 5p
4.
5.
7. 2x 2 5x 3 and 3x 2 7x 4
8. 4d 2 8d 7 and 5d 2 6d 9
6.
7.
9. 2b2 8 and 5b 8
10. 4x 3 and 3x 2 9x
8.
9.
11. 8y3 5y2 and 5y2 2y
12. 9x 4 2x 2 and 2x 2 3
10.
11.
13. 2a 2 4a3 and 3a 3 2a2
14. 9m3 2m and 6m 4m3
12.
13.
15. 4x 2 2 7x and
5 8x 6x 2
16. 5b3 8b 2b2 and
3b2 7b3 5b
14.
15.
16.
Remove the parentheses in each of the following expressions, and simplify when
possible.
17. (2a 3b)
18. (7x 4y)
17.
18.
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19.
19. 5a (2b 3c)
20. 7x (4y 3z)
20.
21.
21. 9r (3r 5s)
22. 10m (3m 2n)
22.
23.
23. 5p (3p 2q)
24. 8d (7c 2d)
24.
277
ANSWERS
25.
Subtract.
26.
25. x 4 from 2x 3
26. x 2 from 3x 5
27. 3m2 2m from 4m2 5m
28. 9a2 5a from 11a 2 10a
29.
29. 6y 2 5y from 4y2 5y
30. 9n2 4n from 7n2 4n
30.
31. x 2 4x 3 from 3x 2 5x 2
32. 3x 2 2x 4 from 5x 2 8x 3
33. 3a 7 from 8a2 9a
34. 3x 3 x 2 from 4x 3 5x
33.
35. 4b2 3b from 5b 2b2
36. 7y 3y 2 from 3y2 2y
34.
37. x2 5 8x from
38. 4x 2x 2 4x3 from
27.
28.
31.
32.
3x 8x 7
2
4x 3 x 3x 2
35.
36.
Perform the indicated operations.
37.
39. Subtract 3b 2 from the sum of 4b 2 and 5b 3.
38.
40. Subtract 5m 7 from the sum of 2m 8 and 9m 2.
39.
41. Subtract 3x 2 2x 1 from the sum of x 2 5x 2 and 2x 2 7x 8.
40.
42. Subtract 4x 2 5x 3 from the sum of x 2 3x 7 and 2x 2 2x 9.
41.
42.
43. Subtract 2x 2 3x from the sum of 4x 2 5 and 2x 7.
43.
44. Subtract 5a 2 3a from the sum of 3a 3 and 5a 2 5.
44.
45.
46.
46. Subtract the sum of 7r 3 4r2 and 3r 3 + 4r 2 from 2r 3 + 3r 2.
47.
Add, using the vertical method.
48.
47. 2w 2 + 7, 3w 5, and 4w 2 5w
49.
48. 3x 2 4x 2, 6x 3, and 2x 2 8
50.
49. 3x 2 3x 4, 4x 2 3x 3, and 2x 2 x 7
50. 5x 2 2x 4, x 2 2x 3, and 2x 2 4x 3
278
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45. Subtract the sum of 3y2 3y and 5y2 3y from 2y2 8y.
ANSWERS
51.
Subtract, using the vertical method.
52.
51. 3a 2 2a from 5a 2 3a
52. 6r3 4r2 from 4r 3 2r 2
53.
53. 5x 2 6x 7 from 8x 2 5x 7
54. 8x 2 4x 2 from 9x 2 8x 6
54.
55.
55. 5x 2 3x from 8x 2 9
56. 7x 2 6x from 9x 2 3
56.
57.
Perform the indicated operations.
58.
57. [(9x 3x 5) (3x 2x 1)] (x 2x 3)
2
2
2
59.
58. [(5x 2 2x 3) (2x 2 x 2)] (2x 2 3x 5)
60.
Find values for a, b, c, and d so that the following equations are true.
61.
59. 3ax4 5x3 x 2 cx 2 9x4 bx 3 x 2 2d
62.
63.
60. (4ax3 3bx2 10) 3(x3 4x2 cx d) x2 6x 8
64.
61. Geometry. A rectangle has sides of 8x 9 and 6x 7. Find the polynomial that
represents its perimeter.
6x 7
8x 9
62. Geometry. A triangle has sides 3x 7, 4x 9, and 5x 6. Find the polynomial
5x
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7
3x 6
that represents its perimeter.
4x 9
63. Business. The cost of producing x units of an item is C 150 25x. The revenue
for selling x units is R 90x x 2. The profit is given by the revenue minus the cost.
Find the polynomial that represents profit.
64. Business. The revenue for selling y units is R 3y2 2y 5 and the cost of
producing y units is C y2 y 3. Find the polynomial that represents profit.
279
ANSWERS
a.
Getting Ready for Section 3.4 [Section 1.7]
b.
Multiply.
c.
(a) x5 x7
(b) y8 y12
(c) 2a 3 d 4
d.
(d) 3m5 m2
(e) 4r5 3r
(f) 6w 2 5w3
e.
(g) (2x2)(8x7)
(h) (10a)(3a5)
f.
Answers
g.
1. 9a 4
3. 13b2 18b
5. 2x2
7. 5x2 2x 1
3
3
3
9. 2b 5b 16
11. 8y 2y
13. a 4a2
15. 2x2 x 3
17. 2a 3b
19. 5a 2b 3c
21. 6r 5s
23. 8p 2q
25. x 7
27. m2 3m
29. 2y2
31. 2x2 x 1
33. 8a2 12a 7
35. 6b2 8b
37. 2x2 12
39. 6b 1
2
41. 10x 9
43. 2x 5x 12
45. 6y2 8y
47. 6w2 2w 2
49. 9x2 x
51. 2a2 5a
53. 3x2 x
55. 3x2 3x 9
2
57. 5x 3x 9
59. a 3, b 5, c 0, d 1
61. 28x 4
63. x2 65x 150
a. x12
b. y20
c. 2a7
d. 3m7
e. 12r6
5
9
6
f. 30w
g. 16x
h. 30a
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h.
280