Download Chapter 4 Polynomials and Exponents

C
H
A
P
T
E
R
4
Polynomials and
Exponents
he nineteenth-century physician and physicist, Jean Louis Marie
Poiseuille (1799–1869) is given credit for discovering a formula
associated with the circulation of blood through arteries.
Poiseuille’s law, as it is known, can be used to determine the velocity of blood in an artery at a given distance from the center of the
artery. The formula states that the flow of blood in an artery is faster toward the center of the blood vessel and is slower toward the outside.
Blood flow can also be affected by a person’s blood pressure, the length
of the blood vessel, and the viscosity of the blood itself.
In later years, Poiseuille’s continued interest in blood circulation
led him to experiments to show that blood pressure rises and falls
when a person exhales and inhales. In modern medicine, physicians
can use Poiseuille’s law to determine how much the radius of a
blocked blood vessel must be widened to create a healthy flow of
blood.
In this chapter you will study polynomials, the fundamental
expressions of algebra. Polynomials are to algebra what integers are to arithmetic. We use polynomials to represent
quantities in general, such as perimeter, area, revenue,
and the volume of blood flowing through an artery. In
Exercise 85 of Section 4.4, you will see Poiseuille’s
law represented by a polynomial.
T
r
|
▲
▲
R
|
e-Text Main Menu
|
Textbook Table of Contents
208
(4-2)
Chapter 4
Polynomials and Exponents
4.1
ADDITION AND SUBTRACTION
OF POLYNOMIALS
We first used polynomials in Chapter 1 but did not identify them as polynomials.
Polynomials also occurred in the equations and inequalities of Chapter 2. In this
section we will define polynomials and begin a thorough study of polynomials.
In this
section
●
Polynomials
Polynomials
●
Value of a Polynomial
●
Addition of Polynomials
In Chapter 1 we defined a term as an expression containing a number or the product
of a number and one or more variables raised to powers. Some examples of terms are
●
Subtraction of Polynomials
●
Applications
4x 3, x 2y 3, 6ab, and 2.
A polynomial is a single term or a finite sum of terms. The powers of the variables
in a polynomial must be positive integers. For example,
4x 3 (15x 2) x (2)
is a polynomial. Because it is simpler to write addition of a negative as subtraction,
this polynomial is usually written as
4x 3 15x 2 x 2.
study
The degree of a polynomial in one variable is the highest power of the variable in
the polynomial. So 4x 3 15x 2 x 2 has degree 3 and 7w w 2 has degree 2.
The degree of a term is the power of the variable in the term. Because the last term
has no variable, its degree is 0.
tip
Many commuting students
find it difficult to get help. Students are often stuck on a
minor point that can be easily
resolved over the telephone.
So ask your instructor if he or
she will answer questions over
the telephone and during
what hours.
Thirddegree
term
Seconddegree
term
Firstdegree
term
Zerodegree
term
A single number is called a constant and so the last term is the constant term. The
degree of a polynomial consisting of a single number such as 8 is 0.
The number preceding the variable in each term is called the coefficient of that
variable or the coefficient of that term. In 4x 3 15x 2 x 2 the coefficient of x 3
is 4, the coefficient of x 2 is 15, and the coefficient of x is 1 because x 1 x.
1
Identifying coefficients
Determine the coefficients of x 3 and x 2 in each polynomial:
b) 4x 6 x 3 x
a) x 3 5x 2 6
Solution
a) Write the polynomial as 1 x 3 5x 2 6 to see that the coefficient of x 3
is 1 and the coefficient of x 2 is 5.
b) The x 2-term is missing in 4x 6 x 3 x. Because 4x 6 x 3 x can be written as
4x 6 1 x 3 0 x 2 x,
the coefficient of x 3 is 1 and the coefficient of x 2 is 0.
■
For simplicity we generally write polynomials with the exponents decreasing
from left to right and the constant term last. So we write
x 3 4x 2 5x 1
rather than
4x 2 1 5x x 3.
When a polynomial is written with decreasing exponents, the coefficient of the first
term is called the leading coefficient.
|
▲
▲
E X A M P L E
4x3 15x2 x 2
|
e-Text Main Menu
|
Textbook Table of Contents
4.1
Addition and Subtraction of Polynomials
(4-3)
209
Certain polynomials are given special names. A monomial is a polynomial that
has one term, a binomial is a polynomial that has two terms, and a trinomial is a
polynomial that has three terms. For example, 3x5 is a monomial, 2x 1 is a binomial, and 4x 6 3x 2 is a trinomial.
E X A M P L E
study
2
Types of polynomials
Identify each polynomial as a monomial, binomial, or trinomial and state its degree.
b) x 43 x 2
c) 5x
d) 12
a) 5x 2 7x 3 2
tip
Solution
a) The polynomial 5x 2 7x 3 2 is a third-degree trinomial.
b) The polynomial x 43 x 2 is a binomial with degree 43.
c) Because 5x 5x1, this polynomial is a monomial with degree 1.
d) The polynomial 12 is a monomial with degree 0.
Be active in class. Don’t be embarrassed to ask questions or
answer questions. You can
often learn more from a
wrong answer than a right
one. Your instructor knows
that you are not yet an expert
in algebra. Instructors love
active classrooms and they
will not think less of you for
speaking out.
E X A M P L E
■
Value of a Polynomial
A polynomial is an algebraic expression. Like other algebraic expressions involving
variables, a polynomial has no specific value unless the variables are replaced by
numbers. A polynomial can be evaluated with or without the function notation
discussed in Chapter 3.
3
Evaluating polynomials
a) Find the value of 3x4 x 3 20x 3 when x 1.
b) Find the value of 3x4 x 3 20x 3 when x 2.
c) If P(x) 3x 4 x 3 20x 3, find P(1).
calculator
Solution
a) Replace x by 1 in the polynomial:
3x4 x 3 20x 3 3(1)4 (1)3 20(1) 3
3 1 20 3
19
So the value of the polynomial is 19 when x 1.
b) Replace x by 2 in the polynomial:
close-up
To evaluate the polynomial in
Example 3 with a calculator,
first use Y to define the
polynomial.
3x4 x3 20x 3 3(2)4 (2)3 20(2) 3
3(16) (8) 40 3
48 8 40 3
77
So the value of the polynomial is 77 when x 2.
c) This is a repeat of part (a) using the function notation from Chapter 3. P(1), read
“P of 1,” is the value of the polynomial P(x) when x is 1. To find P(1), replace x
by 1 in the formula for P(x):
Then find y1(2) and y1(1).
P(x) 3x 4 x 3 20x 3
P(1) 3(1)4 (1)3 20(1) 3
19
|
▲
▲
So P(1) 19. The value of the polynomial when x 1 is 19.
|
e-Text Main Menu
|
Textbook Table of Contents
■
210
(4-4)
Chapter 4
Polynomials and Exponents
Addition of Polynomials
You learned how to combine like terms in Chapter 1. Also, you combined like terms
when solving equations in Chapter 2. Addition of polynomials is done simply by
adding the like terms.
Addition of Polynomials
To add two polynomials, add the like terms.
Polynomials can be added horizontally or vertically, as shown in the next
example.
E X A M P L E
helpful
4
Adding polynomials
Perform the indicated operation.
a) (x 2 6x 5) (3x 2 5x 9)
b) (5a3 3a 7) (4a2 3a 7)
Solution
a) We can use the commutative and associative properties to get the like terms next
to each other and then combine them:
hint
(x 2 6x 5) (3x 2 5x 9) x 2 3x 2 6x 5x 5 9
2x 2 x 4
b) When adding vertically, we line up the like terms:
When we perform operations
with polynomials and write
the results as equations, those
equations are identities. For
example,
3a 7
4a 3a 7
3
5a 4a2
5a3
(x 1) (3x 5) 4x 6
2
is an identity. This equation is
satisfied by every real number.
■
Add.
Subtraction of Polynomials
When we subtract polynomials, we subtract the like terms. Because a b a (b), we can subtract by adding the opposite of the second polynomial to the
first polynomial. Remember that a negative sign in front of parentheses changes the
sign of each term in the parentheses. For example,
(x 2 2x 8) x 2 2x 8.
Polynomials can be subtracted horizontally or vertically, as shown in the next
example.
5
Subtracting polynomials
Perform the indicated operation.
a) (x 2 5x 3) (4x 2 8x 9)
b) (4y 3 3y 2) (5y 2 7y 6)
Solution
a) (x 2 5x 3) (4x 2 8x 9) x 2 5x 3 4x 2 8x 9
3x 2 13x 6
|
▲
▲
E X A M P L E
|
e-Text Main Menu
|
Textbook Table of Contents
Change signs.
Add.
4.1
helpful
Addition and Subtraction of Polynomials
(4-5)
211
b) To subtract 5y 2 7y 6 from 4y 3 3y 2 vertically, we line up the like
terms as we do for addition:
hint
For subtraction, write the original problem and then rewrite
it as addition with the signs
changed. Many students have
trouble when they write the
original problem and then
overwrite the signs. Vertical
subtraction is essential for performing long division of polynomials in Section 4.5.
4y 3
3y 2
(5y 2 7y 6)
Now change the signs of 5y 2 7y 6 and add the like terms:
3y 2
4y 3
2
5y 7y 6
3
4y 5y2 4y 8
■
CAUTION
When adding or subtracting polynomials vertically, be sure to
line up the like terms.
In the next example we combine addition and subtraction of polynomials.
E X A M P L E
6
Adding and subtracting
Perform the indicated operations:
(2x 2 3x) (x 3 6) (x4 6x 2 9)
Solution
Remove the parentheses and combine the like terms:
(2x 2 3x) (x 3 6) (x4 6x 2 9) 2x 2 3x x 3 6 x4 6x 2 9
■
x4 x 3 8x 2 3x 15
Applications
Polynomials are often used to represent unknown quantities. In certain situations it
is necessary to add or subtract such polynomials.
E X A M P L E
7
Profit from prints
Trey pays $60 per day for a permit to sell famous art prints in the Student Union
Mall. Each print costs him $4, so the polynomial 4x 60 represents his daily cost
in dollars for x prints sold. He sells the prints for $10 each. So the polynomial 10x
represents his daily revenue for x prints sold. Find a polynomial that represents his
daily profit from selling x prints. Evaluate the profit polynomial for x 30.
Solution
Because profit is revenue minus cost, we can subtract the corresponding polynomials to get a polynomial that represents the daily profit:
10x (4x 60) 10x 4x 60
6x 60
His daily profit from selling x prints is 6x 60 dollars. Evaluate this profit polynomial for x 30:
6x 60 6(30) 60 120
|
▲
▲
So if Trey sells 30 prints, his profit is $120.
|
e-Text Main Menu
|
Textbook Table of Contents
■
212
(4-6)
Chapter 4
Polynomials and Exponents
M A T H
A T
W O R K
The message we hear today about healthy eating is
“more fiber and less fat.” But healthy eating can be
challenging. Jill Brown, Registered Dietitian and
owner of Healthy Habits Nutritional Counseling
and Consulting, provides nutritional counseling for
NUTRITIONIST
people who are basically healthy but interested in
improving their eating habits.
On a client’s first visit, an extensive nutrition, medical, and family history is
taken. Behavior patterns are discussed, and nutritional goals are set. Ms. Brown
strives for a realistic rather than an idealistic approach. Whenever possible, a client
should eat foods that are high in vitamin content rather than take vitamin pills.
Moreover, certain frame types and family history make a slight and slender look difficult to achieve. Here, Ms. Brown might use the Harris-Benedict formula to determine the daily basal energy expenditure based on age, gender, and size. In addition
to diet, exercise is discussed and encouraged. Finally, Ms. Brown provides a support system and serves as a “nutritional coach.” By following her advice, many of
her clients lower their blood pressure, reduce their cholesterol, have more energy,
and look and feel better.
In Exercises 101 and 102 of this section you will use the Harris-Benedict
formula to calculate the basal energy expenditure for a female and a male.
WARM-UPS
True or false? Explain your answer.
In the polynomial 2x 2 4x 7 the coefficient of x is 4. False
The degree of the polynomial x 2 5x 9x 3 6 is 2. False
In the polynomial x 2 x the coefficient of x is 1. True
The degree of the polynomial x 2 x is 2. True
A binomial always has a degree of 2. False
The polynomial 3x 2 5x 9 is a trinomial. True
Every trinomial has degree 2. False
x 2 7x 2 6x 2 for any value of x. True
(3x 2 8x 6) (x 2 4x 9) 4x 2 4x 3 for any value of x.
True
10. (x 2 4x) (x 2 3x) 7x for any value of x. False
1.
2.
3.
4.
5.
6.
7.
8.
9.
4. 1
EXERCISES
|
▲
▲
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a term?
A term is a single number or the product of a number and
one or more variables raised to powers.
|
e-Text Main Menu
2. What is a polynomial?
A polynomial is a single term or a finite sum of terms.
3. What is the degree of a polynomial?
The degree of a polynomial in one variable is the highest
power of the variable in the polynomial.
|
Textbook Table of Contents
4.1
Determine the coefficients of x 3 and x 2 in each polynomial. See
Example 1.
7. 3x 3 7x 2
8. 10x 3 x 2 10, 1
9. x4 6x 2 9 0, 6
10. x 5 x 3 3
1, 0
3
2
x
7x
x3
x2
1
1
11. 4
12. 2x 1
3
2
2
4
Identify each polynomial as a monomial, binomial, or trinomial
and state its degree. See Example 2.
13. 1
14. 5
15. m 3
Monomial, 0
Monomial, 0
Monomial, 3
8
16. 3a
17. 4x 7
18. a 6
Monomial, 8
Binomial, 1
Binomial, 1
10
2
6
3
19. x 3x 2
20. y 6y 9
Trinomial, 10
Trinomial, 6
6
21. x 1
22. b2 4
Binomial, 6
Binomial, 2
3
2
23. a a 5
24. x 2 4x 9
Trinomial, 3
Trinomial, 2
60. (345.56x 347.4) (56.6x 433)
Add or subtract the polynomials as indicated. See Examples 4
and 5.
61. Add:
62. Add:
3a 4
2w 8
a6
w3
Evaluate each polynomial as indicated. See Example 3.
25. Evaluate 2x 2 3x 1 for x 1. 6
26. Evaluate 3x 2 x 2 for x 2. 16
27. Evaluate 3x 3 x 2 3x 4 for x 3.
85
28. Evaluate 2x 4 3x 2 5x 9 for x 2.
43
29. If P(x) 3x 4 2x 3 7, find P(2). 71
30. If P(x) 2x 3 5x 2 12, find P(5).
137
31. If P(x) 1.2x 3 4.3x 2.4, find P(1.45).
4.97665
32. If P(x) 3.5x 4 4.6x 3 5.5, find P(2.36).
▲
▲
Perform the indicated operation. See Example 4.
33. (x 3) (3x 5)
34. (x 2) (x 3)
4x 8
2x 1
35. (q 3) (q 3)
36. (q 4) (q 6)
2q
2q 10
37. (3x 2) (x2 4)
38. (5x 2 2) (3x 2 1)
x x 2
2x
3
39. (4x 1) (x 3 5x 6) x
9x 7
40. (3x 7) (x 2 4x 6) x
x 1
2
2
41. (a 3a 1) (2a 4a 5) 3a
7a 4
42. (w 2 2w 1) (2w 5 w2) 2w
4
43. (w 2 9w 3) (w 4w 2 8)
3w
8w 5
44. (a 3 a2 5a) (6 a 3a2 ) a
4a
6
6
2
2
45. (5.76x 3.14x 7.09) (3.9x 1.21x 5.6)
9.66x
1.93x 1.49
|
e-Text Main Menu
213
46. (8.5x 2 3.27x 9.33) (x 2 4.39x 2.32)
9.5
x 11.65
Perform the indicated operation. See Example 5.
47. (x 2) (5x 8)
48. (x 7) (3x 1)
4x 6
2x
49. (m 2) (m 3)
50. (m 5) (m 9)
5
4
51. (2z 2 3z) (3z 2 5z) 52. (z 2 4z) (5z 2 3z)
z
2z
53. (w 5 w 3) (w4 w2) w
w
54. (w 6 w 3) (w 2 w) w
w
55. (t 2 3t 4) (t 2 5t 9) 2t
t
56. (t 2 6t 7) (5t 2 3t 2)
57. (9 3y y 2) (2 5y y 2 )
58. (4 5y y 3) (2 3y y 2 ) y
59. (3.55x 879) (26.4x 455.8)
4. What is the value of a polynomial?
The value of a polynomial is the number obtained when the
variable is replaced by a number.
5. How do we add polynomials?
Polynomials are added by adding the like terms.
6. How do we subtract polynomials?
Polynomials are subtracted by subtracting like terms.
|
(4-7)
Addition and Subtraction of Polynomials
4a 2
63. Subtract:
3x 11
5x 7
3w 5
64. Subtract:
4x 3
2x 9
2x
4
65. Add:
ab
ab
2x 6
66. Add:
s6
s1
2a
67. Subtract:
3m 1
2m 6
2s 7
68. Subtract:
5n 2
3n 4
m 7
69. Add:
2x 2 x 3
2x 2 x 4
8n 6
70. Add:
x 2 4x 6
3x 2 x 5
4x
1
71. Subtract:
3a3 5a2 7
2a3 4a2 2a
2x
3x 11
72. Subtract:
2b3 7b2
9
b3
4b 2
73. Subtract:
x 2 3x 6
x2
3
a
b
b
4b
74. Subtract:
x4 3x 2 2
3x4 2x 2
3
75. Add:
y 3 4y 2 6y 5
y 3 3y 2 2y 9
2
x
2
76. Add:
q 2 4q 9
3q 2 7q 5
2y
|
y
7
14
Textbook Table of Contents
q
q
14
7
214
(4-8)
Chapter 4
Polynomials and Exponents
Perform the operations indicated.
77. Find the sum of 2m 9 and 3m 4.
78. Find the sum of 3n 2 and 6m 3.
79. Find the difference when 7y 3 is subtracted from
9y 2. 2y 1
80. Find the difference when 2y 1 is subtracted from
3y 4. 5y 3
81. Subtract x 2 3x 1 from the sum of 2x 2 x 3 and
x 2 5x 9. 2x
7x 5
82. Subtract 2y 2 3y 8 from the sum of 3y 2 2y 6
and 7y 2 8y 3. 6y
3y 11
Perform the indicated operations. See Example 6.
83. (4m 2) (2m 4) (9m 1)
m 3
84. (5m 6) (8m 3) (5m 3) 8m 12
85. (6y 2) (8y 3) (9y 2)
11y 3
86. (5y 1) (8y 4) (y 3)
14y
c) Find the cost of labor for 500 transmissions and the cost
of materials for 500 transmissions.
a) 0.4x
450x 1350 b) $326,350
c) $275,550, $50,800
93. Perimeter of a triangle. The shortest side of a triangle is x
meters, and the other two sides are 3x 1 and 2x 4 meters. Write a polynomial that represents the perimeter and
then evaluate the perimeter polynomial if x is 4 meters.
6x 3 meters, 27 meters
94. Perimeter of a rectangle. The width of a rectangular playground is 2x 5 feet, and the length is 3x 9 feet. Write
a polynomial that represents the perimeter and then evaluate this perimeter polynomial if x is 4 feet.
10x 8 feet, 48 feet
87. (x 2 5x 4) (6x 2 8x 9) (3x 2 7x 1)
2x
6
88. (8x 2 5x 12) (3x 2 9x 18)
(3x 2 9x 4)
13
10
89. (6z4 3z 3 7z 2) (5z 3 3z 2 2)
(z4 z 2 5)
5z
8z
3z
7
90. (v 3 v2 1) (v4 v2 v 1) (v3 3v2 6)
6
Solve each problem. See Example 7.
91. Profitable pumps. Walter Waterman, of Walter’s Water
Pumps in Winnipeg has found that when he produces x
water pumps per month, his revenue is x 2 400x 300
dollars. His cost for producing x water pumps per month is
x 2 300x 200 dollars. Write a polynomial that represents his monthly profit for x water pumps. Evaluate this
profit polynomial for x 50.
100x 500 dollars, $5500
92. Manufacturing costs. Ace manufacturing has determined
that the cost of labor for producing x transmissions is
0.3x 2 400x 550 dollars, while the cost of materials is
0.1x 2 50x 800 dollars.
a) Write a polynomial that represents the total cost of materials and labor for producing x transmissions.
b) Evaluate the total cost polynomial for x 500.
2x 5 ft
3x 9 ft
FIGURE FOR EXERCISE 94
95. Before and after. Jessica traveled 2x 50 miles in the
morning and 3x 10 miles in the afternoon. Write a polynomial that represents the total distance that she traveled.
Find the total distance if x 20.
5x 40 miles, 140 miles
96. Total distance. Hanson drove his rig at x mph for 3 hours,
then increased his speed to x 15 mph and drove for 2
more hours. Write a polynomial that represents the total distance that he traveled. Find the total distance if x 45 mph.
5x 30 miles, 255 miles
97. Sky divers. Bob and Betty simultaneously jump from two
airplanes at different altitudes. Bob’s altitude t seconds
after leaving the plane is 16t 2 6600 feet. Betty’s altitude t seconds after leaving the plane is 16t 2 7400
feet. Write a polynomial that represents the difference between their altitudes t seconds after leaving the planes.
What is the difference between their altitudes 3 seconds
after leaving the planes? 800 feet, 800 feet
Total
0.5
16t 2 7400 ft
16t 2 6600 ft
Labor
Materials
0
0
500
1000
Number of transmissions
FIGURE FOR EXERCISE 97
FIGURE FOR EXERCISE 92
|
▲
▲
Cost (millions of dollars)
1
|
e-Text Main Menu
|
Textbook Table of Contents
4.2
98. Height difference. A red ball and a green ball are simultaneously tossed into the air. The red ball is given an initial velocity of 96 feet per second, and its height t seconds
after it is tossed is 16t 2 96t feet. The green ball is
given an initial velocity of 30 feet per second, and its
height t seconds after it is tossed is 16t 2 80t feet.
a) Find a polynomial that represents the difference in the
heights of the two balls. 16t feet
b) How much higher is the red ball 2 seconds after the
balls are tossed? 32 feet
c) In reality, when does the difference in the heights stop
increasing? when green ball hits ground
Multiplication of Polynomials
(4-9)
215
101. Harris-Benedict for females. The Harris-Benedict polynomial
655.1 9.56w 1.85h 4.68a
represents the number of calories needed to maintain a
female at rest for 24 hours, where w is her weight in
kilograms, h is her height in centimeters, and a is her
age. Find the number of calories needed by a 30-year-old
54-kilogram female who is 157 centimeters tall.
1321.39 calories
102. Harris-Benedict for males. The Harris-Benedict polynomial
66.5 13.75w 5.0h 6.78a
Height (feet)
150
represents the number of calories needed to maintain a
male at rest for 24 hours, where w is his weight in kilograms, h is his height in centimeters, and a is his age. Find
the number of calories needed by a 40-year-old 90-kilogram male who is 185 centimeters tall.
1957.8 calories
Red ball
Green ball
100
50
GET TING MORE INVOLVED
Difference
0
0
1
2
3
Time (seconds)
103. Discussion. Is the sum of two natural numbers always
a natural number? Is the sum of two integers always an integer? Is the sum of two polynomials always a polynomial? Explain.
Yes, yes, yes
104. Discussion. Is the difference of two natural numbers always a natural number? Is the difference of two rational
numbers always a rational number? Is the difference of
two polynomials always a polynomial? Explain.
No, yes, yes
105. Writing. Explain why the polynomial 24 7x3 5x2 x
has degree 3 and not degree 4.
106. Discussion. Which of the following polynomials does
not have degree 2? Explain.
a) r 2 b) 2 4 c) y 2 4 d) x 2 x 4
e) a2 3a 9
b and d
4
FIGURE FOR EXERCISE 98
99. Total interest. Donald received 0.08(x 554) dollars
interest on one investment and 0.09(x 335) interest on
another investment. Write a polynomial that represents
the total interest he received. What is the total interest if
x 1000?
0.17x 74.47 dollars, $244.47
100. Total acid. Deborah figured that the amount of acid in
one bottle of solution is 0.12x milliliters and the amount
of acid in another bottle of solution is 0.22(75 x) milliliters. Find a polynomial that represents the total amount
of acid? What is the total amount of acid if x 50?
x 16.5 milliliters, 11.5 milliliters
In this
4.2
MULTIPLICATION OF POLYNOMIALS
section
●
Multiplying Monomials with
the Product Rule
●
Multiplying Polynomials
●
The Opposite of a
Polynomial
●
Applications
You learned to multiply some polynomials in Chapter 1. In this section you will
learn how to multiply any two polynomials.
Multiplying Monomials with the Product Rule
To multiply two monomials, such as x3 and x5, recall that
|
▲
▲
x3 x x x
|
e-Text Main Menu
|
and
x 5 x x x x x.
Textbook Table of Contents
216
(4-10)
Chapter 4
Polynomials and Exponents
So
3 factors
5 factors
x 3 x 5 (x x x )(x x x x x) x8.
8 factors
3
The exponent of the product of x and x5 is the sum of the exponents 3 and 5. This
example illustrates the product rule for multiplying exponential expressions.
Product Rule
If a is any real number and m and n are any positive integers, then
am an amn.
E X A M P L E
study
1
Multiplying monomials
Find the indicated products.
b) (2ab)(3ab)
a) x2 x4 x
c) 4x2y2 3xy5
d) (3a)2
Solution
a) x2 x4 x x2 x4 x1
x7 Product rule
b) (2ab)(3ab) (2)(3) a a b b
6a2b2 Product rule
c) (4x2y2)(3xy5) (4)(3)x2 x y2 y5
12x3y7 Product rule
d) (3a)2 3a 3a
9a2
tip
As soon as possible after class,
find a quiet place and work on
your homework. The longer
you wait the harder it is to remember what happened in
class.
■
CAUTION
Be sure to distinguish between adding and multiplying monomials. You can add like terms to get 3x4 2x4 5x4, but you cannot combine
the terms in 3w5 6w2. However, you can multiply any two monomials:
3x4 2x4 6x8 and 3w5 6w2 18w7.
Multiplying Polynomials
To multiply a monomial and a polynomial, we use the distributive property.
E X A M P L E
study
2
Multiplying monomials and polynomials
Find each product.
b) (y2 3y 4)(2y)
a) 3x2(x3 4x)
tip
Solution
a) 3x2(x3 4x) 3x2 x3 3x2 4x
3x5 12x3
Distributive property
b) (y2 3y 4)(2y) y2(2y) 3y (2y) 4(2y)
2y3 (6y2) (8y)
2y3 6y2 8y
|
▲
▲
When doing homework or taking notes, use a pencil with an
eraser. Everyone makes mistakes. If you get a problem
wrong, don’t start over. Check
your work for errors and use
the eraser.It is better to find out
where you went wrong than
to simply get the right answer.
c) a(b c)
|
e-Text Main Menu
|
Textbook Table of Contents
Distributive property
4.2
c) a(b c) (a)b (a)c
ab ac
ac ab
(4-11)
Multiplication of Polynomials
217
Distributive property
Note in part (c) that either of the last two binomials is the correct answer. The last
■
one is just a little simpler to read.
Just as we use the distributive property to find the product of a monomial and a
polynomial, we can use the distributive property to find the product of two binomials and the product of a binomial and a trinomial.
E X A M P L E
3
Multiplying polynomials
Use the distributive property to find each product.
a) (x 2)(x 5)
b) (x 3)(x2 2x 7)
Solution
a) First multiply each term of x 5 by x 2:
(x 2)(x 5) (x 2)x (x 2)5
x2 2x 5x 10
x2 7x 10
Distributive property
Distributive property
Combine like terms.
b) First multiply each term of the trinomial by x 3:
(x 3)(x2 2x 7) (x 3)x2 (x 3)2x (x 3)(7)
x3 3x2 2x2 6x 7x 21
x 5x x 21
3
2
Distributive
property
Distributive property
Combine like terms.
■
Products of polynomials can also be found by arranging the multiplication
vertically like multiplication of whole numbers.
E X A M P L E
helpful
4
Multiplying vertically
Find each product.
a) (x 2)(3x 7)
hint
Solution
a)
3x 7
x2
6x 14
2
3x 7x
3x2 x 14
Many students find vertical
multiplication easier than applying the distributive property twice horizontally. However, you should learn both
methods because horizontal
multiplication will help you
with factoring by grouping in
Section 5.2.
b) (x y)(a 3)
xy
a3
3x 3y
b)
← 2 times 3x 7
← x times 3x 7
Add.
ax ay
ax ay 3x 3y
|
▲
▲
These examples illustrate the following rule.
|
e-Text Main Menu
|
Textbook Table of Contents
■
218
(4-12)
Chapter 4
Polynomials and Exponents
Multiplication of Polynomials
To multiply polynomials, multiply each term of one polynomial by every
term of the other polynomial, then combine like terms.
The Opposite of a Polynomial
Note the result of multiplying the difference a b by 1:
1(a b) a b b a
Because multiplying by 1 is the same as taking the opposite, we can write
(a b) b a.
So a b and b a are opposites or additive inverses of each other. If a and b are
replaced by numbers, the values of a b and b a are additive inverses. For example, 3 7 4 and 7 3 4.
CAUTION
E X A M P L E
5
The opposite of a b is a b, not a b.
Opposite of a polynomial
Find the opposite of each polynomial.
a) x 2
b) 9 y2
c) a 4
d) x 2 6x 3
Solution
a) (x 2) 2 x
b) (9 y2 ) y2 9
c) (a 4) a 4
d) (x 2 6x 3) x2 6x 3
■
Applications
E X A M P L E
6
Multiplying polynomials
A parking lot is 20 yards wide and 30 yards long. If the college increases the length
and width by the same amount to handle an increasing number of cars, then what
polynomial represents the area of the new lot? What is the new area if the increase
is 15 yards?
Solution
If x is the amount of increase, then the new lot will be x 20 yards wide
and x 30 yards long as shown in Fig. 4.1. Multiply the length and width to get the
area:
x
(x 20)(x 30) (x 20)x (x 20)30
x 2 20x 30x 600
x 2 50x 600
30 yd
The polynomial x 2 50x 600 represents the area of the new lot. If x 15, then
20 yd
x 2 50x 600 (15)2 50(15) 600 1575.
FIGURE 4.1
If the increase is 15 yards, then the area of the lot will be 1575 square yards.
|
▲
▲
x
|
e-Text Main Menu
|
Textbook Table of Contents
■
4.2
WARM-UPS
Multiplication of Polynomials
(4-13)
219
True or false? Explain your answer.
3x3 5x4 15x12 for any value of x.
1.
2.
3.
4.
5.
6.
7.
False
3x 2x 5x for any value of x. False
(3y3)2 9y6 for any value of y. True
3x(5x 7x2) 15x3 21x2 for any value of x. False
2x(x2 3x 4) 2x3 6x2 8x for any number x. True
2(3 x) 2x 6 for any number x. True
(a b)(c d) ac ad bc bd for any values of a, b, c, and d.
True
8. (x 7) 7 x for any value of x. True
9. 83 37 (37 83) True
10. The opposite of x 3 is x 3 for any number x. False
4. 2
2
7
9
EXERCISES
19. (5y)2
25y
21. (2x3)2
4x
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is the product rule for exponents?
2. Why is the sum of two monomials not necessarily a monomial?
The sum of two monomials can be a binomial if the terms
are not like terms.
3. What property of the real numbers is used when multiplying a monomial and a polynomial?
To multiply a monomial and a polynomial we use the
distributive property.
4. What property of the real numbers is used when multiplying two binomials?
To multiply two binomials we use the distributive property
twice.
5. How do we multiply any two polynomials?
To multiply any two polynomials we multiply each term
of the first polynomial by every term of the second
polynomial.
6. How do we find the opposite of a polynomial?
To find the opposite of a polynomial, change the sign of
each term in the polynomial.
|
▲
▲
Find each product. See Example 1.
7. 3x2 9x3
8. 5x7 3x5
27x
15x
10. 3y12 5y15
11. 6x2 5x2
15y
x
13. (9x10)(3x7)
14. (2x2)(8x9)
27x
16x
16. 12sq 3s
17. 3wt 8w7t6
qs
24t
|
Find each product. See Example 2.
23. 4y2(y5 2y)
6t3(t5 3t2)
3y(6y 4)
9y(y2 1)
(y2 5y 6)(3y)
3
(x3 5x2 1)7x2 7x
x(y2 x2)
ab(a2 b2)
(3ab3 a2b2 2a3b)5a
(3c2d d3 1)8cd2 24
1
33. t2v(4t3v2 6tv 4v)
2
1
34. m2n3(6mn2 3mn 12)
3
24.
25.
26.
27.
28.
29.
30.
31.
32.
Use the distributive property to find each product. See
Example 3.
9. 2a3 7a8
14a
12. 2x 2 8x 5
16x
15. 6st 9st
54s
18. h8k3 5h
5h
e-Text Main Menu
20. (6x)2
36x
22. (3y5)2
9y
35. (x 1)(x 2)
x
3x 2
37. (x 3)(x 5)
x
2x 15
39. (t 4)(t 9)
t
13t 36
41. (x 1)(x2 2x 2)
x
3x
4x
|
36. (x 6)(x 3)
x
9x 18
38. ( y 2)( y 4)
y
y 8
40. (w 3)(w 5)
w
8w 15
42. (x 1)(x2 x 1)
x
Textbook Table of Contents
220
(4-14)
Chapter 4
Polynomials and Exponents
43. (3y 2)(2y2 y 3)
44. (4y 3)(y2 3y 1)
6y
7
6
4y
15y
13
3
45. (y2z 2y4)(y2z 3z2 y4)
2y
3
y
3y
3
2
4 2
46. (m 4mn )(6m n 3m6 m2n4)
18m
23m
3m
4m
Find each product vertically. See Example 4.
47. 2a 3
48. 2w 6
a5
w5
2a
7a
49. 7x 30
2x 5
15
14x
95x
51. 5x 2
4x 3
20x
7x
53. m 3n
2a b
2w
4w
50. 5x 7
3x 6
150
30
15x
51x
52. 4x 3
2x 6
6
8x
18x
54. 3x 7
a 2b
2am 6an mb
55. x2 3x 2
x6
42
18
3ax
7a
6xb
56. x2 3x 5
x7
14b
x
10x
26
58. 3x2 5x 2
5x 6
35
15x
7x
60. a2 b2
a2 b2
12
59. x y
xy
x
y
61. x2 xy y2
xy
a
b
62. 4w2 2wv v2
2w v
9x
16x
57. 2a3 3a2 4
2a 3
a
3nb
83. (5x 6)(5x 6)
25x
36
85. 2x2(3x5 4x2)
6x
8x
87. (m 1)(m2 m 1)
m
1
89. (3x 2)(x2 x 9)
90. (5 6y)(3y2 y 7)
12
a
12
20x
|
▲
▲
x
y
8w
v
Find the opposite of each polynomial. See Example 5.
63. 3t u u 3t
64. 3t u 3t u
65. 3x y
3x y
66. x 3y 3y x
67. 3a2 a 6
6
2
68. 3b b 6
6
69. 3v2 v 6
6
70. 3t2 t 6
Perform the indicated operation.
71. 3x(2x 9)
72. 1(2 3x)
x
27x
3x 2
73. 2 3x(2x 9)
74. 6 3(4x 8)
27x 2
30 12x
75. (2 3x) (2x 9)
76. (2 3x) (2x 9)
5
11
6 2
3 2
77. (6x ) 36x
78. (3a b)
79. 3ab3(2a2b7)
a
80. 4xst 8xs
81. (5x 6)(5x 6)
82. (5x 6)(5x 6)
25x
60x 36
25x
60x 36
|
e-Text Main Menu
84. (2x 9)(2x 9)
4x
81
86. 4a3(3ab3 2ab3)
4a b
88. (a b)(a2 ab b2)
a
b
Solve each problem. See Example 6.
91. Office space. The length of a professor’s office is x feet,
and the width is x 4 feet. Write a polynomial that represents the area. Find the area if x 10 ft.
x
4x square feet, 140 square feet
92. Swimming space. The length of a rectangular swimming
pool is 2x 1 meters, and the width is x 2 meters. Write
a polynomial that represents the area. Find the area if x is
5 meters. 2x
3x 2 square meters, 63 square meters
93. Area of a truss. A roof truss is in the shape of a triangle with
a height of x feet and a base of 2x 1 feet. Write a polynomial that represents the area of the triangle. What is the area
if x is 5 feet?
1
x ft
2x 1 ft
FIGURE FOR EXERCISE 93
94. Volume of a box. The length, width, and height of a box
are x, 2x, and 3x 5 inches, respectively. Write a polynomial that represents its volume. 6x
10x
3x 5
2x
x
FIGURE FOR EXERCISE 94
95. Number pairs. If two numbers differ by 5, then what polynomial represents their product? x
5x
96. Number pairs. If two numbers have a sum of 9, then what
polynomial represents their product? 9x x
97. Area of a rectangle. The length of a rectangle is 2.3x 1.2 meters, and its width is 3.5x 5.1 meters. What
polynomial represents its area?
8.05
x
|
Textbook Table of Contents
Total revenue (thousands of dollars)
4.3
98. Patchwork. A quilt patch cut in the shape of a triangle
has a base of 5x inches and a height of 1.732x inches.
What polynomial represents its area?
4.33x2 square inches
Multiplication of Binomials
(4-15)
221
10
9
8
7
6
5
4
3
2
1
0 2
4
6
8 10 12 14 16 18
Price (dollars)
FIGURE FOR EXERCISE 100
101. Periodic deposits. At the beginning of each year for 5
years, an investor invests $10 in a mutual fund with an average annual return of r. If we let x 1 r, then at the
end of the first year (just before the next investment) the
value is 10x dollars. Because $10 is then added to the 10x
dollars, the amount at the end of the second year
is (10x 10)x dollars. Find a polynomial that represents
the value of the investment at the end of the fifth year.
Evaluate this polynomial if r 10%.
1.732 x
5x
102. Increasing deposits. At the beginning of each year for 5
years, an investor invests in a mutual fund with an average
annual return of r. The first year, she invests $10; the second year, she invests $20; the third year, she invests $30;
the fourth year, she invests $40; the fifth year, she invests
$50. Let x 1 r as in Exercise 101 and write a polynomial in x that represents the value of the investment at the
end of the fifth year. Evaluate this polynomial for r 8%.
FIGURE FOR EXERCISE 98
99. Total revenue. If a promoter charges p dollars per ticket
for a concert in Tulsa, then she expects to sell 40,000 1000p tickets to the concert. How many tickets will she sell
if the tickets are $10 each? Find the total revenue when the
tickets are $10 each. What polynomial represents the total
revenue expected for the concert when the tickets are p
dollars each? 30,000, $300,000, 40,000p 1000p
100. Manufacturing shirts. If a manufacturer charges p dollars each for rugby shirts, then he expects to sell
2000 100p shirts per week. What polynomial represents the total revenue expected for a week? How many
shirts will be sold if the manufacturer charges $20 each
for the shirts? Find the total revenue when the shirts are
sold for $20 each. Use the bar graph to determine the
price that will give the maximum total revenue.
2000p 100p
4.3
GET TING MORE INVOLVED
103. Discussion. Name all properties of the real numbers that
are used in finding the following products:
a) 2ab3c2 5a2bc
b) (x2 3)(x2 8x 6)
104. Discussion. Find the product of 27 and 436 without
using a calculator. Then use the distributive property to
find the product (20 7)(400 30 6) as you would
find the product of a binomial and a trinomial. Explain
how the two methods are related.
MULTIPLICATION OF BINOMIALS
In Section 4.2 you learned to multiply polynomials. In this section you will learn a
rule that makes multiplication of binomials simpler.
In this
section
The FOIL Method
●
The FOIL Method
●
Multiplying Binomials
Quickly
We can use the distributive property to find the product of two binomials. For
example,
|
▲
▲
(x 2)(x 3) (x 2)x (x 2)3
x2 2x 3x 6
x2 5x 6
|
e-Text Main Menu
|
Textbook Table of Contents
Distributive property
Distributive property
Combine like terms.
222
(4-16)
Chapter 4
Polynomials and Exponents
There are four terms in x 2 2x 3x 6. The term x2 is the product of the first
term of each binomial, x and x. The term 3x is the product of the two outer terms,
3 and x. The term 2x is the product of the two inner terms, 2 and x. The term 6 is the
product of the last term of each binomial, 2 and 3. We can connect the terms multiplied by lines as follows:
L
F
(x 2)(x 3)
F First terms
O Outer terms
I Inner terms
L Last terms
I
O
If you remember the word FOIL, you can get the product of the two binomials much
faster than writing out all of the steps above. This method is called the FOIL
method. The name should make it easier to remember.
E X A M P L E
1
Using the FOIL method
Find each product.
a) (x 2)(x 4)
c) (a b)(2a b)
b) (2x 5)(3x 4)
d) (x 3)( y 5)
Solution
helpful
L
F
hint
F
O
I
L
a) (x 2)(x 4) x 4x 2x 8
x 2 2x 8 Combine the like terms.
2
You may have to practice FOIL
a while to get good at it. However, the better you are at
FOIL, the easier you will find
factoring in Chapter 5.
I
O
b) (2x 5)(3x 4) 6x2 8x 15x 20
6x2 7x 20 Combine the like terms.
c) (a b)(2a b) 2a2 ab 2ab b2
2a2 3ab b2
d) (x 3)( y 5) xy 5x 3y 15 There are no like terms to combine.
■
FOIL can be used to multiply any two binomials. The binomials in the next
example have higher powers than those of Example 1.
E X A M P L E
study
2
Using the FOIL method
Find each product.
a) (x3 3)(x3 6)
tip
Solution
a) (x3 3)(x3 6) x6 6x3 3x3 18
x6 3x3 18
b) (2a2 1)(a2 5) 2a4 10a2 a2 5
2a4 11a2 5
Remember that everything
we do in solving problems is
based on principles (which are
also called rules, theorems,
and definitions). These principles justify the steps we take.
Be sure that you understand
the reasons. If you just memorize procedures without understanding, you will soon forget the procedures.
■
Multiplying Binomials Quickly
The outer and inner products in the FOIL method are often like terms, and we can
combine them without writing them down. Once you become proficient at using
FOIL, you can find the product of two binomials without writing anything except
the answer.
|
▲
▲
b) (2a2 1)(a2 5)
|
e-Text Main Menu
|
Textbook Table of Contents
4.3
E X A M P L E
3
Combine like terms: 3x 4x 7x.
Combine like terms: 10x x 9x.
■
Combine like terms: 6a 6a 0.
Area of a garden
Sheila has a square garden with sides of length x feet. If she increases the length by
7 feet and decreases the width by 2 feet, then what trinomial represents the area of
the new rectangular garden?
x2
Solution
The length of the new garden is x 7 and the width is x 2 as shown in Fig. 4.2.
■
The area is (x 7)(x 2) or x2 5x 14 square feet.
x7
FIGURE 4.2
@ http://ellerbruch.nmu.edu/cs255/npaquett/binomials.html
WARM-UPS
True or false? Answer true only if the equation is true for all values of
the variable or variables. Explain your answer.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
4. 3
223
4
x
x
(4-17)
Using FOIL to find a product quickly
Find each product. Write down only the answer.
a) (x 3)(x 4)
b) (2x 1)(x 5)
c) (a 6)(a 6)
Solution
a) (x 3)(x 4) x2 7x 12
b) (2x 1)(x 5) 2x2 9x 5
c) (a 6)(a 6) a2 36
E X A M P L E
Multiplication of Binomials
(x 3)(x 2) x2 6 False
(x 2)( y 1) xy x 2y 2 True
(3a 5)(2a 1) 6a2 3a 10a 5 True
(y 3)( y 2) y2 y 6 True
(x2 2)(x2 3) x4 5x2 6 True
(3a2 2)(3a2 2) 9a2 4 False
(t 3)(t 5) t2 8t 15 True
(y 9)( y 2) y2 11y 18 False
(x 4)(x 7) x2 4x 28 False
It is not necessary to learn FOIL as long as you can get the answer.
False
EXERCISES
|
▲
▲
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What property of the real numbers do we usually use to find
the product of two binomials?
We use the distributive property to find the product of two
binomials.
2. What does FOIL stand for?
FOIL stands for first, outer, inner, and last.
3. What is the purpose of FOIL?
The purpose of FOIL is to provide a faster method for finding the product of two binomials.
|
e-Text Main Menu
4. What is the maximum number of terms that can be obtained
when two binomials are multiplied?
The maximum number of terms obtained in multiplying binomials is four.
Use FOIL to find each product. See Example 1.
5. (x 2)(x 4)
6. (x 3)(x 5)
7. (a 3)(a 2)
8. (b 1)(b 2)
9. (2x 1)(x 2)
|
Textbook Table of Contents
224
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
(4-18)
Chapter 4
Polynomials and Exponents
(2y 5)(y 2)
(2a 3)(a 1)
(3x 5)(x 4) 3
(w 50)(w 10) w
(w 30)(w 20) w
(y a)(y 5) y
(a t)(3 y) 3a
(5 w)(w m) 5w
(a h)(b t) ab
(2m 3t)(5m 3t)
(2x 5y)(x y) 2x
(5a 2b)(9a 7b)
(11x 3y)(x 4y)
55.
56.
57.
58.
(h 7w)(h 7w) h
(h 7q)(h 7q) h
(2h2 1)(2h2 1) 4
(3h2 1)(3h2 1) 9
Perform the indicated operations.
1
1
8
59. 2a 4a 2
2
2
1
60. 3b 6b 3
3
1
1 1
1
61. x x 2
3 4
2
62.
Use FOIL to find each product. See Example 2.
23. (x 2 5)(x 2 2)
24. (y2 1)(y 2 2)
25. (h 3 5)(h 3 5)
26. (y6 1)( y6 4)
27. (3b3 2)(b3 4)
28. (5n4 1)(n4 3)
29. (y2 3)( y 2)
30. (x 1)(x2 1) x
x
31. (3m3 n2)(2m3 3n2)
32. (6y4 2z2)(6y4 3z2)
33. (3u 2v 2)(4u2v 6)
34. (5y3w 2 z)(2y3w 2 3z)
|
▲
▲
Find each product. Try to write only the answer. See Example 3.
35. (b 4)(b 5) b
36. (y 8)(y 4)
37. (x 3)(x 9) x
38. (m 7)(m 8)
39. (a 5)(a 5) a
40. (t 4)(t 4) t
41. (2x 1)(2x 1)
42. (3y 4)(3y 4)
43. (z 10)(z 10)
44. (3h 5)(3h 5)
45. (a b)(a b) a
46. (x y)(x y) x
47. (a 1)(a 2) a
48. (b 8)(b 1) b
49. (2x 1)(x 3)
50. (3y 5)(y 3)
51. (5t 2)(t 1) 5
52. (2t 3)(2t 1)
53. (h 7)(h 9) h
54. (h 7w)(h 7w)
|
e-Text Main Menu
63.
64.
65.
66.
67.
68.
69.
70.
3t 42t 2
2
1
1
1
2x 4(3x 1)(2x 5)
4xy3(2x y)(3x y) 24
(x 1)(x 1)(x 3) x
(a 3)(a 4)(a 5) a
(3x 2)(3x 2)(x 5)
(x 6)(9x 4)(9x 4) 81x
(x 1)(x 2) (x 3)(x 4)
(k 4)(k 9) (k 3)(k 7) k
15
Solve each problem.
71. Area of a rug. Find a trinomial that represents the area of
a rectangular rug whose sides are x 3 feet and
2x 1 feet.
2x
5x 3 square feet
x+3
2x – 1
FIGURE FOR EXERCISE 71
72. Area of a parallelogram. Find a trinomial that represents
the area of a parallelogram whose base is 3x 2 meters
and whose height is 2x 3 meters.
6x
13x 6 square meters
73. Area of a sail. The sail of a tall ship is triangular in shape
with a base of 4.57x 3 meters and a height of
2.3x 1.33 meters. Find a polynomial that represents the
area of the triangle.
5.2555x
0.41095x 1.995 square meters
74. Area of a square. A square has a side of length 1.732x 1.414 meters. Find a polynomial that represents its area.
2.9998x
4.898
1.999 square meters
|
Textbook Table of Contents
4.4
(4-19)
Special Products
225
GET TING MORE INVOLVED
76. Exploration. Find the area of each of the four regions
shown in the figure. What is the total area of the four
regions? What does this exercise illustrate?
75. Exploration. Find the area of each of the four regions
shown in the figure. What is the total area of the four regions? What does this exercise illustrate?
12 ft h
h ft
h
7h 12 ft
(h 3)(h 4) h
7h 12
h
ft
4 ft
a
b
h ft
h ft
b
b
3 ft
3 ft
a
a
h
ft
4 ft
a
FIGURE FOR EXERCISE 75
4.4
In this
section
●
The Square of a Binomial
●
Product of a Sum and a
Difference
●
Higher Powers of Binomials
●
Applications to Area
helpful
SPECIAL PRODUCTS
In Section 4.3 you learned the FOIL method to make multiplying binomials
simpler. In this section you will learn rules for squaring binomials and for finding the product of a sum and a difference. These products are called special
products.
To compute (a b)2, the square of a binomial, we can write it as (a b)(a b)
and use FOIL:
(a b)2 (a b)(a b)
a2 ab ab b2
a2 2ab b2
To visualize the square of a
sum, draw a square with sides
of length a b as shown.
b
a
a2
ab
b
ab
b2
FIGURE FOR EXERCISE 76
The Square of a Binomial
hint
a
b
So to square a b, we square the first term (a2), add twice the product of the two
terms (2ab), then add the square of the last term (b2). The square of a binomial
occurs so frequently that it is helpful to learn this new rule to find it. The rule for
squaring a sum is given symbolically as follows.
The Square of a Sum
The area of the large square is
(a b)2. It comes from four
terms as stated in the rule for
the square of a sum.
1
|
Using the rule for squaring a sum
Find the square of each sum.
a) (x 3)2
▲
▲
E X A M P L E
(a b)2 a2 2ab b2
|
e-Text Main Menu
|
b) (2a 5)2
Textbook Table of Contents
(4-20)
helpful
Chapter 4
Polynomials and Exponents
Solution
a) (x 3)2 x2 2(x)(3) 32 x 2 6x 9
hint
↑
Square
of
first
You can mentally find the
square of a number that ends
in 5. To find 252 take the 2
(from 25), multiply it by 3 (the
next integer) to get 6, and
then annex 25 for 625. To find
352, find 3 4 12 and annex
25 for 1225. To see why this
works, write
(30 5)2
302 2 30 5 52
30(30 2 5) 25
30 40 25
1200 25
1225.
2
Now find 45 and 552 mentally.
226
↑
Twice
the
product
↑
Square
of
last
b) (2a 5)2 (2a)2 2(2a)(5) 52
4a2 20a 25
■
CAUTION
Do not forget the middle term when squaring a sum. The
equation (x 3)2 x2 6x 9 is an identity, but (x 3)2 x2 9 is not an
identity. For example, if x 1 in (x 3)2 x2 9, then we get 42 12 9,
which is false.
When we use FOIL to find (a b)2, we see that
(a b)2 (a b)(a b)
a2 ab ab b2
a2 2ab b2.
So to square a b, we square the first term (a2), subtract twice the product of the
two terms (2ab), and add the square of the last term (b2). The rule for squaring a
difference is given symbolically as follows.
The Square of a Difference
(a b)2 a2 2ab b2
E X A M P L E
helpful
2
Using the rule for squaring a difference
Find the square of each difference.
a) (x 4)2
b) (4b 5y)2
Solution
a) (x 4)2 x2 2(x)(4) 42
x2 8x 16
b) (4b 5y)2 (4b)2 2(4b)(5y) (5y)2
16b2 40by 25y2
hint
Many students keep using
FOIL to find the square of a
sum or difference. However,
learning the new rules for
these special cases will pay off
in the future.
■
Product of a Sum and a Difference
If we multiply the sum a b and the difference a b by using FOIL, we get
(a b)(a b) a2 ab ab b2
a2 b2.
|
▲
▲
The inner and outer products have a sum of 0. So the product of a sum and a difference of the same two terms is equal to the difference of two squares.
|
e-Text Main Menu
|
Textbook Table of Contents
4.4
Special Products
(4-21)
227
The Product of a Sum and a Difference
(a b)(a b) a2 b2
E X A M P L E
helpful
3
Product of a sum and a difference
Find each product.
a) (x 2)(x 2)
b) (b 7)(b 7)
Solution
a) (x 2)(x 2) x2 4
b) (b 7)(b 7) b2 49
c) (3x 5)(3x 5) 9x2 25
hint
You can use
(a b)(a b) a2 b2
to perform mental arithmetic
tricks like
19 21 (20 1)(20 1)
400 1
399
What is 29 31? 28 32?
E X A M P L E
study
c) (3x 5)(3x 5)
■
Higher Powers of Binomials
To find a power of a binomial that is higher than 2, we can use the rule for squaring
a binomial along with the method of multiplying binomials using the distributive
property. Finding the second or higher power of a binomial is called expanding the
binomial because the result has more terms than the original.
4
Higher powers of a binomial
Expand each binomial.
a) (x 4)3
b) (y 2)4
Solution
a) (x 4)3 (x 4)2(x 4)
(x 2 8x 16)(x 4)
(x 2 8x 16)x (x 2 8x 16)4
x3 8x 2 16x 4x2 32x 64
x3 12x2 48x 64
b) (y 2)4 (y 2)2(y 2)2
(y2 4y 4)(y2 4y 4)
(y2 4y 4)(y2) (y2 4y 4)(4y) (y2 4y 4)(4)
y4 4y3 4y2 4y3 16y2 16y 4y2 16y 16
■
y4 8y3 24y2 32y 16
tip
Correct answers often have
more than one form. If your
answer to an exercise doesn’t
agree with the one in the back
of this text, try to determine if
it is simply a different form of
the answer. For example, 1x
2
and x look different but they
2
are equivalent expressions.
Applications to Area
E X A M P L E
5
Area of a pizza
A pizza parlor saves money by making all of its round pizzas one inch smaller in radius than advertised. Write a trinomial for the actual area of a pizza with an advertised radius of r inches.
|
▲
▲
Solution
A pizza advertised as r inches has an actual radius of r 1 inches. The actual area
is (r 1)2:
(r 1)2 (r 2 2r 1) r 2 2r .
■
So r 2 2r is a trinomial representing the actual area.
|
e-Text Main Menu
|
Textbook Table of Contents
228
(4-22)
Chapter 4
Polynomials and Exponents
WARM-UPS
True or false? Explain your answer.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
4. 4
(2 3)2 22 32 False
(x 3)2 x 2 6x 9 for any value of x. True
(3 5)2 9 25 30 True
(2x 7)2 4x 2 28x 49 for any value of x. True
(y 8)2 y2 64 for any value of y. False
The product of a sum and a difference of the same two terms is equal to the
difference of two squares. True
(40 1)(40 1) 1599 True
49 51 2499 True
(x 3)2 x2 3x 9 for any value of x. False
The square of a sum is equal to a sum of two squares. False
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What are the special products?
2. What is the rule for squaring a sum?
3. Why do we need a new rule to find the square of a sum
when we already have FOIL?
It is faster to do by the new rule than with FOIL.
4. What happens to the inner and outer products in the product
of a sum and a difference?
In (a b)(
) the inner and outer products have a sum
of zero.
5. What is the rule for finding the product of a sum and a
difference?
6. How can you find higher powers of binomials?
Higher powers of binomials are found by using the distributive property.
|
▲
▲
Square each binomial. See Example 1.
7. (x 1)2
8. (y 2)2
x
2x 1
y
4
4
10. (z 3)2
9. (y 4)2
y
8
z
6z 9
12. (2m 7)2
11. (3x 8)2
9x
48
64
4m
28m
14. (x z)2
13. (s t)2
s
2st
x
2xz
15. (2x y)2
16. (3t v)2
4x
4xy
9t
6tv
18. (3z 5k)2
17. (2t 3h)2
4t
12
9z
30kz
|
49
e-Text Main Menu
Square each binomial. See Example 2.
19. (a 3)2
20. (w 4)2
a
6a 9
w
8w 16
2
21. (t 1)
22. (t 6)2
t
t 1
t
t 36
2
24. (5a 6)2
23. (3t 2)
9t
12t 4
25a
60a 36
2
2
26. (r w)
25. (s t)
s
2st
r
rw
2
27. (3a b)
28. (4w 7)2
9a
6ab
16w
56w
2
29. (3z 5y)
30. (2z 3w)2
9z
30yz
4z
12wz
Find each product. See Example 3.
31. (a 5)(a 5)
32. (x 6)(x 6)
a
25
x
36
33. (y 1)(y 1)
34. (p 2)(p 2)
y
1
p
4
35. (3x 8)(3x 8)
36. (6x 1)(6x 1)
9x
36x
1
37. (r s)(r s)
38. (b y)(b y)
r
s
b
y
39. (8y 3a)(8y 3a)
40. (4u 9v)(4u 9v)
64y
9a
16u
81v
42. (3y2 1)(3y2 1)
41. (5x 2 2)(5x 2 2)
25x
4
9y
1
Expand each binomial. See Example 4.
43. (x 1)3
44. (y 1)3
45. (2a 3)3
46. (3w 1)3
47. (a 3)4
|
Textbook Table of Contents
4.4
48. (2b 1)4 16
49. (a b)4 a
50. (2a 3b)4
Find each product.
51. (a 20)(a 20)
a
400
53. (x 8)(x 7)
x
15x 56
55. (4x 1)(4x 1)
16x
1
57. (9y 1)2
81y
18y 1
59. (2t 5)(3t 4)
6t
7t
61. (2t 5)2
4t
20t 25
63. (2t 5)(2t 5)
4t
25
2
65. (x 1)(x 2 1)
x
1
3
67. (2y 9)2
4y
36y
81
3
2 2
69. (2x 3y )
4x
12x
9y
1
1
71. x 2
3
t + 3 cm
2
v k(R r)(R r),
where k is a constant. Rewrite the formula using a special
product rule. v k(R
r
R
In Exercises 81–90, solve each problem.
81. Shrinking garden. Rose’s garden is a square with sides of
length x feet. Next spring she plans to make it rectangular
by lengthening one side 5 feet and shortening the other side
by 5 feet. What polynomial represents the new area? By how
much will the area of the new garden differ from that of the
old garden? x
25 square feet, 25 square feet smaller
▲
▲
t cm
85. Poiseuille’s law. According to the nineteenth-century physician Poiseuille, the velocity (in centimeters per second) of
blood r centimeters from the center of an artery of radius R
centimeters is given by
73. (0.2x 0.1)2
0.04x
0.04x 0.01
74. (0.1y 0.5)2
0.01y
0.1y 0.25
75. (a b)3
a
a
3
76. (2a 3b)3
8a
36a
77. (1.5x 3.8)2
2.25x
11.4x 14.44
78. (3.45a 2.3)2
11.9025a
15.87a 5.29
79. (3.5t 2.5)(3.5t 2.5)
12.25t
6.25
80. (4.5h 5.7)(4.5h 5.7)
20.25h
32.49
|
229
FIGURE FOR EXERCISE 84
2
1
72. y 3
2
(4-23)
82. Square lot. Sam lives on a lot that he thought was a
square, 157 feet by 157 feet. When he had it surveyed, he
discovered that one side was actually 2 feet longer than
he thought and the other was actually 2 feet shorter than he
thought. How much less area does he have than he thought
he had? 4 square feet
83. Area of a circle. Find a polynomial that represents the area
of a circle whose radius is b 1 meters. Use the value
3.14 for . 3.14b
6.28b 3.14 square meters
84. Comparing dart boards. A toy store sells two sizes of circular dartboards. The larger of the two has a radius that is
3 inches greater than that of the other. The radius of the
smaller dartboard is t inches. Find a polynomial that represents the difference in area between the two dartboards.
6t 9 square centimeters
52. (1 x)(1 x)
1 x
54. (x 9)(x 5)
x
x 45
56. (9y 1)(9y 1)
81y
1
58. (4x 1)2
16x
8x 1
60. (2t 5)(3t 4)
6t
t 20
62. (2t 5)2
4t
20t 25
64. (3t 4)(3t 4)
9t
16
3
66. (y 1)(y3 1)
y
1
4
68. (3z 8)2
9z
48z
64
5
3 2
70. (4y 2w )
2
Special Products
|
e-Text Main Menu
r
FIGURE FOR EXERCISE 85
86. Going in circles. A promoter is planning a circular race
track with an inside radius of r feet and a width of w feet.
|
Textbook Table of Contents
(4-24)
Chapter 4
Polynomials and Exponents
89. Investing in treasury bills. An investment advisor uses the
polynomial P(1 r)10 to predict the value in 10 years of a
client’s investment of P dollars with an average annual return r. The accompanying graph shows historic average annual returns for the last 20 years for various asset classes
(T. Rowe Price, www.troweprice.com). Use the historical
average return to predict the value in 10 years of an investment of $10,000 in U.S. treasury bills? $20,230.06
w
r
FIGURE FOR EXERCISE 86
The cost in dollars for paving the track is given by the
formula
C 1.2[(r w)2 r 2].
Use a special product rule to simplify this formula. What is
the cost of paving the track if the inside radius is 1000 feet
and the width of the track is 40 feet?
C 1.2 [2rw w
87. Compounded annually. P dollars is invested at annual interest rate r for 2 years. If the interest is compounded annually, then the polynomial P(1 r)2 represents the value of
the investment after 2 years. Rewrite this expression without parentheses. Evaluate the polynomial if P $200 and
r 10%.
Pr Pr 2
88. Compounded semiannually. P dollars is invested at annual interest rate r for 1 year. If the interest is compounded
2
semiannually, then the polynomial P1 represents
r
2
the value of the investment after 1 year. Rewrite this expression without parentheses. Evaluate the polynomial if
P $200 and r 10%.
Pr
P Pr
4.5
Average annual return (percent)
230
20
16
16.7%
12
10.3%
7.3%
8
3.4%
4
0
Large Long-term U.S. Inflation
company corporate treasury
stocks
bonds
bills
FIGURE FOR EXERCISES 89 AND 90
90. Comparing investments. How much more would the investment in Exercise 89 be worth in 10 years if the client
invests in large company stocks rather than U.S. treasury
bills? $26,619.83
GET TING MORE INVOLVED
91. Writing. What is the difference between the equations
(x 5)2 x 2 10x 25 and (x 5)2 x2 25?
The first is an identity and the second is a conditional
equation.
92. Writing. Is it possible to square a sum or a difference without using the rules presented in this section? Why should
you learn the rules given in this section?
A sum or difference can be squared with the distributive
property, FOIL, or the special product rules. It is easier with
the special product rules.
DIVISION OF POLYNOMIALS
You multiplied polynomials in Section 4.2. In this section you will learn to divide
polynomials.
In this
section
●
Dividing Monomials Using the Quotient Rule
Dividing Monomials Using
the Quotient Rule
●
Dividing a Polynomial by a
Monomial
●
Dividing a Polynomial by a
Binomial
In Chapter 1 we used the definition of division to divide signed numbers. Because
the definition of division applies to any division, we restate it here.
Division of Real Numbers
If a, b, and c are any numbers with b 0, then
|
▲
▲
abc
|
e-Text Main Menu
|
provided that
c b a.
Textbook Table of Contents
4.5
study
Division of Polynomials
(4-25)
231
If a b c, we call a the dividend, b the divisor, and c (or a b) the quotient.
You can find the quotient of two monomials by writing the quotient as a fraction
and then reducing the fraction. For example,
tip
Establish a regular routine of
eating, sleeping, and exercise.
The ability to concentrate
depends on adequate sleep,
decent nutrition,and the physical well-being that comes
with exercise.
x5
x x x x x
x 5 x 2 2 x 3.
x
x x
You can be sure that x 3 is correct by checking that x 3 x 2 x 5. You can also divide
x 2 by x 5, but the result is not a monomial:
1
x2
1 x x
x 2 x 5 5 3
x
x x x x x x
Note that the exponent 3 can be obtained in either case by subtracting 5 and 2.
These examples illustrate the quotient rule for exponents.
Quotient Rule
Suppose a 0, and m and n are positive integers.
am
If m n, then n a mn.
a
am
1
If n m, then n .
a
a nm
Note that if you use the quotient rule to subtract the exponents in x 4 x 4, you
get the expression x 44, or x 0, which has not been defined yet. Because we must
have x 4 x 4 1 if x 0, we define the zero power of a nonzero real number to
be 1. We do not define the expression 00.
Zero Exponent
For any nonzero real number a,
a0 1.
E X A M P L E
1
Using the definition of zero exponent
Simplify each expression. Assume that all variables are nonzero real numbers.
b) (3xy)0
c) a0 b0
a) 50
Solution
a) 50 1
b) (3xy)0 1
c) a0 b0 1 1 2
■
With the definition of zero exponent the quotient rule is valid for all positive integers as stated.
2
|
Using the quotient rule in dividing monomials
Find each quotient.
y9
12b2
a) 5
b) c) 6x3 (2x9)
y
3b7
▲
▲
E X A M P L E
|
e-Text Main Menu
|
Textbook Table of Contents
x8y2
d) x2y2
232
(4-26)
Chapter 4
helpful
Polynomials and Exponents
Solution
y9
a) 5 y95 y4
y
Use the definition of division to check that y4 y5 y9.
12b2 12 b2
1
4
b) 7 7 4 72 5
3b
b
b
3 b
Use the definition of division to check that
4
12b7
12b2.
5 3b7 b
b5
hint
Recall that the order of operations gives multiplication
and division an equal ranking
and says to do them in order
from left to right. So without
parentheses,
6x 3 2x 9
actually means
6x 3 9
x.
2
6x 3 3
c) 6x 3 (2x 9) 2x 9
x6
Use the definition of division to check that
3
6x 9
9
6x 3.
2x
x6
x6
x8y2
x 8 y2
d) x 6 y0 x 6
x 2 y2 x 2 y2
Use the definition of division to check that x 6 x 2 y 2 x 8y2.
■
We showed more steps in Example 2 than are necessary. For division problems
like these you should try to write down only the quotient.
Dividing a Polynomial by a Monomial
We divided some simple polynomials by monomials in Chapter 1. For example,
6x 8 1
(6x 8) 3x 4.
2
2
We use the distributive property to take one-half of 6x and one-half of 8 to get
3x 4. So both 6x and 8 are divided by 2. To divide any polynomial by a monomial, we divide each term of the polynomial by the monomial.
E X A M P L E
study
3
Dividing a polynomial by a monomial
Find the quotient for (8x6 12x4 4x 2) (4x 2).
tip
Solution
8x6 12x 4 4x 2 8x6 12x 4 4x 2
2
4x 2
4x 2
4x 2
4x
4
2
2x 3x 1
The quotient is 2x 4 3x 2 1. We can check by multiplying.
4x 2(2x 4 3x 2 1) 8x 6 12x 4 4x 2.
■
Because division by zero is undefined, we will always assume that the divisor
is nonzero in any quotient involving variables. For example, the division in Example 3 is valid only if 4x2 0, or x 0.
|
▲
▲
Play offensive math,not defensive math. A student who says,
“Give me a question and I’ll see
if I can answer it,” is playing
defensive math. The student
is taking a passive approach
to learning. A student who
takes an active approach and
knows the usual questions
and answers for each topic is
playing offensive math.
|
e-Text Main Menu
|
Textbook Table of Contents
4.5
Division of Polynomials
(4-27)
233
Dividing a Polynomial by a Binomial
Division of whole numbers is often done with a procedure called long division. For
example, 253 is divided by 7 as follows:
36 ← Quotient
Divisor → 72
53 ← Dividend
21
43
42
1 ← Remainder
Note that 36 7 1 253. It is always true that
(quotient)(divisor) (remainder) dividend.
To divide a polynomial by a binomial, we perform the division like long division of whole numbers. For example, to divide x2 3x 10 by x 2, we get the
first term of the quotient by dividing the first term of x 2 into the first term of
x 2 3x 10. So divide x 2 by x to get x, then multiply and subtract as follows:
1 Divide:
2 Multiply:
3
Subtract:
x
x 2x2
x
3
0
1
x2 2x
5x
x2 x x
x (x 2) x2 2x
3x 2x 5x
Now bring down 10 and continue the process. We get the second term of the quotient (below) by dividing the first term of x 2 into the first term of 5x 10. So
divide 5x by x to get 5:
1 Divide:
2 Multiply:
3
Subtract:
x5
x 2x2
x
3
0
1
x2 2x ↓
5x 10
5x 10
0
5x x 5
Bring down 10.
5(x 2) 5x 10
10 (10) 0
So the quotient is x 5, and the remainder is 0.
In the next example we must rearrange the dividend before dividing.
E X A M P L E
helpful
4
Dividing a polynomial by a binomial
Divide 2x 3 4 7x 2 by 2x 3, and identify the quotient and the remainder.
Solution
Rearrange the dividend as 2x3 7x2 4. Because the x-term in the dividend is
missing, we write 0 x for it:
hint
x2 2x 3
2x 32
x 3
x
72
0x
4
3
2x 3x2
4x 2 0 x
4x 2 6x
6x 4
6x 9
13
|
▲
▲
Students usually have the
most difficulty with the subtraction part of long division.
So pay particular attention to
that step and double check
your work.
|
e-Text Main Menu
|
2x 3 (2x) x 2
7x 2 (3x2) 4x 2
0 x 6x 6x
4 (9) 13
Textbook Table of Contents
234
(4-28)
Chapter 4
Polynomials and Exponents
The quotient is x2 2x 3, and the remainder is 13. Note that the degree of the
remainder is 0 and the degree of the divisor is 1. To check, we must verify that
(2x 3)(x 2 2x 3) 13 2x3 7x2 4.
■
CAUTION
To avoid errors, always write the terms of the divisor and the
dividend in descending order of the exponents and insert a zero for any term that is
missing.
If we divide both sides of the equation
dividend (quotient)(divisor) (remainder)
by the divisor, we get the equation
dividend
remainder
quotient .
divisor
divisor
This fact is used in expressing improper fractions as mixed numbers. For example,
if 19 is divided by 5, the quotient is 3 and the remainder is 4. So
4
19
4
3 3.
5
5
5
We can also use this form to rewrite algebraic fractions.
E X A M P L E
5
Rewriting algebraic fractions
3x
Express in the form
x2
remainder
quotient .
divisor
Solution
Use long division to get the quotient and remainder:
3
3x
0
x 2
3x 6
6
Because the quotient is 3 and the remainder is 6, we can write
6
3x
3 .
x2
x2
■
To check, we must verify that 3(x 2) 6 3x.
CAUTION
When dividing polynomials by long division, we do not stop
until the remainder is 0 or the degree of the remainder is smaller than the degree of
the divisor. For example, we stop dividing in Example 5 because the degree of the
remainder 6 is 0 and the degree of the divisor x 2 is 1.
WARM-UPS
True or false? Explain your answer.
1. y10 y2 y5 for any nonzero value of y. False
2.
7x 2
7
3.
7x 2
7
x 2 for any value of x. False
x 2 for any value of x. True
|
▲
▲
4. If 3x2 6 is divided by 3, the quotient is x2 6.
|
e-Text Main Menu
|
False
Textbook Table of Contents
4.5
WARM-UPS
Division of Polynomials
(4-29)
235
(continued)
5. If 4y2 6y is divided by 2y, the quotient is 2y 3. True
6. The quotient times the remainder plus the dividend equals the divisor.
False
7. (x 2)(x 1) 3 x2 3x 5 for any value of x. True
8. If x2 3x 5 is divided by x 2, then the quotient is x 1. True
9. If x2 3x 5 is divided by x 2, the remainder is 3. True
10. If the remainder is zero, then (divisor)(quotient) dividend. True
@ http://www.sosmath.com/algebra/factor/fac01/fac01.html
4. 5
EXERCISES
9x 2y2
25. 3x 5y2
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What rule is important for dividing monomials?
The quotient rule is used for dividing monomials.
2. What is the meaning of a zero exponent?
The zero power of a nonzero real number is 1.
3. How many terms should you get when dividing a polynomial by a monomial?
When dividing a polynomial by a monomial the quotient
should have the same number of terms as the polynomial.
4. How should the terms of the polynomials be written when
dividing with long division?
The terms of a ploynomial should be written in descending
order of the exponents.
5. How do you know when to stop the process in long division
of polynomials?
The long division process stops when the degree of the remainder is less than the degree of the divisor.
6. How do you handle missing terms in the dividend polynomial when doing long division?
Insert a term with zero coefficient for each missing term
when doing long division.
Simplify each expression. See Example 1.
7. 90 1
8. m0 1
9. (2x3)0 1
3 0
0
0
10. (5a b) 1
11. 2 5 3 1 12. 4 0 8 0
2
0
13. (2x y) 1 14. (a2 b2)0 1
Find the quotients. See Example 3.
3x 6
27. x 2
3
5y 10
28. 2
5
x5 3x4 x3
x
3
29. x2
6y6 9y4 12y2
30. 2y
3
3y2
8x2y2 4x 2y 2xy2
31. 4
2xy
9ab2 6a3b3
32. 3
a
3ab2
33. (x2y3 3x3y2) (x2y) y
34. (4h5k 6h2k2) (2h2k)
Find the quotient and remainder for each division. Check by
using the fact that dividend (divisor)(quotient) remainder.
See Example 4.
35. (x2 5x 13) (x 3) x 2, 7
36. (x2 3x 6) (x 3) x, 6
37. (2x) (x 5) 2, 10
38. (5x) (x 1) 5, 5
39. (a3 4a 3) (a 2) a
2a 8, 13
40. (w3 2w2 3) (w 2) w
4w 8, 13
41. (x2 3x) (x 1) x 4, 4
42. (3x2) (x 1) 3x 3, 3
43. (h3 27) (h 3) h
3h 9, 0
44. (w3 1) (w 1) w
w 1, 0
45. (6x2 13x 7) (3x 2) 2x 3, 1
46. (4b2 25b 3) (4b 1) b 6, 9
47. (x3 x2 x 2) (x 1) x
1, 1
48. (a3 3a2 4a 4) (a 2) a
2, 0
Find each quotient. Try to write only the answer. See Example 2.
x8
15. 2 x
x
6a7
17. 2a12
y9
16. 3 y
y
30b2 1
18. 3b6
19. 12x5 (3x9)
4
x
20. 6y5 (3y10)
21. 6y2 (6y)
y
22. 3a2b (3ab)
4h2k4
24. 3
2hk
3x
|
▲
▲
6x3y2
23. 2x 2y2
|
y
a
2hk
e-Text Main Menu
12z 4y2
26. 2z10y 2
|
Textbook Table of Contents
236
(4-30)
Chapter 4
Polynomials and Exponents
Write each expression in the form
remainder
quotient .
divisor
See Example 5.
2x
3x
49. 50. x5
x1
x
3x
51. 52. x3
x1
a5
x1
53. 1
54. 1
x
a
2y 1
3x 1
55. 56. x
y
x2
x2
57. x
58. x
x1
x1
x2 4
59. x2
x2 1
60. x1
x3
61. x2
x3 1
62. x 1
x1
x3 3
2x2 4
63. 64. x
x
2x
Find each quotient.
65. 6a3b (2a2b)
3a
Solve each problem.
85. Area of a rectangle. The area of a rectangular billboard
is x 2 x 30 square meters. If the length is x 6 meters, find a binomial that represents the width.
x 5 meters
?
GAS FOR LESS
NEXT EXIT
TEXACO
x 6 meters
FIGURE FOR EXERCISE 85
86. Perimeter of a rectangle. The perimeter of a rectangular
backyard is 6x 6 yards. If the width is x yards, find a binomial that represents the length.
2x 3 yards
66. 14x7 (7x2) 2x
x yards
67. 8w4t7 (2w9t3)
68. 9y7z4 (3y3z11) ?
(3a 12) (3)
(6z 3z 2) (3z) 2 z
(3x 2 9x) (3x) x 3
(5x 3 15x 2 25x) (5x) x
3x 5
4
3
2
2
(12x 4x 6x ) (2x )
x
2x 3
(9x 3 3x 2 15x) (3x) 3x
5
(t 2 5t 36) (t 9) t 4
(b2 2b 35) (b 5) b 7
(6w2 7w 5) (3w 5) 2w 1
(4z2 23z 6) (4z 1) z 6
(8x 3 27) (2x 3) 4x
6x 9
(8y 3 1) (2y 1) 4y
2y 1
(t 3 3t 2 5t 6) (t 2) t
t 3
(2u3 13u2 8u 7) (u 7) 2u
u 1
(6v2 4 9v v 3) (v 4)
(14y 8y2 y3 12) (6 y)
|
▲
▲
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
|
e-Text Main Menu
FIGURE FOR EXERCISE 86
GET TING MORE INVOLVED
87. Exploration. Divide x3 1 by x 1, x4 1 by x 1,
and x5 1 by x 1. What is the quotient when x9 1 is
divided by x 1?
88. Exploration. Divide a3 b3 by a b and a4 b4 by
a b. What is the quotient when a8 b8 is divided by
a b?
x
89. Discussion. Are the expressions 150
, 10x 5x, and (10x) x
(5x) equivalent? Before you answer, review the order of
operations in Section 1.5 and evaluate each expression for
x 3.
10x 5x is not equivalent to the other two.
|
Textbook Table of Contents
4.6
4.6
In this
section
Positive Integral Exponents
237
POSITIVE INTEGRAL EXPONENTS
The product rule for positive integral exponents was presented in Section 4.2, and
the quotient rule was presented in Section 4.5. In this section we review those rules
and then further investigate the properties of exponents.
The Product and Quotient
Rules
The Product and Quotient Rules
●
Raising an Exponential
Expression to a Power
The rules that we have already discussed are summarized below.
●
Power of a Product
●
Power of a Quotient
●
Summary of Rules
●
(4-31)
The following rules hold for nonnegative integers m and n and a 0.
a m a n a mn
am
n a mn if m n
a
am
1
if n m
n nm
a
a
a0 1
Product rule
Quotient rule
Zero exponent
CAUTION
The product and quotient rules apply only if the bases of the
expressions are identical. For example, 32 34 36, but the product rule cannot be
applied to 52 34. Note also that the bases are not multiplied: 32 34 96.
Note that in the quotient rule the exponents are always subtracted, as in
x7
x4
x3
and
y5
1
8 3.
y
y
If the larger exponent is in the denominator, then the result is placed in the
denominator.
E X A M P L E
1
Using the product and quotient rules
Use the rules of exponents to simplify each expression. Assume that all variables
represent nonzero real numbers.
b) (3x)0(5x2)(4x)
a) 23 22
2
8x
(3a2b)b9
c) 5
d) 2x
(6a5)a3b2
|
▲
▲
Solution
a) Because the bases are both 2, we can use the product rule:
Product rule
23 22 25
32
Simplify.
0
2
2
b) (3x) (5x )(4x) 1 5x 4x Definition of zero exponent
20x 3
Product rule
2
8x
4
c) 5 3
Quotient rule
2x
x
|
e-Text Main Menu
|
Textbook Table of Contents
238
(4-32)
Chapter 4
Polynomials and Exponents
d) First use the product rule to simplify the numerator and denominator:
study
tip
(3a2b)b9
3a2b10
5 3 2 (6a )a b
6a8b2
Keep track of your time for
one entire week. Account for
how you spend every half
hour. Add up your totals for
sleep, study, work, and recreation. You should be sleeping
50–60 hours per week and
studying 1–2 hours for every
hour you spend in the classroom.
b8
6
2a
Product rule
Quotient rule
■
Raising an Exponential Expression to a Power
When we raise an exponential expression to a power, we can use the product rule to
find the result, as shown in the following example:
(w4)3 w4 w4 w4
w12
Three factors of w4 because of the exponent 3
Product rule
By the product rule we add the three 4’s to get 12, but 12 is also the product of 4 and 3.
This example illustrates the power rule for exponents.
Power Rule
If m and n are nonnegative integers and a 0, then
(a m)n a mn.
In the next example we use the new rule along with the other rules.
2
Using the power rule
Use the rules of exponents to simplify each expression. Assume that all variables
represent nonzero real numbers.
(23)4 27
3(x 5)4
b) c) a) 3x 2(x 3)5
5
9
2 2
15x 22
Solution
a) 3x 2(x 3)5 3x 2x15
3x17
3 4
7
212 27
(2 ) 2
b) 214
25 2 9
219
214
25
32
5 4
3(x )
3x 20
1
c) 22 15x
15x 22 5x 2
Power rule
Product rule
Power rule and product rule
Product rule
Quotient rule
Evaluate 25.
■
Power of a Product
Consider an example of raising a monomial to a power. We will use known rules to
rewrite the expression.
(2x)3 2x 2x 2x
222xxx
23x3
|
▲
▲
E X A M P L E
|
e-Text Main Menu
|
Definition of exponent 3
Commutative and associative properties
Definition of exponents
Textbook Table of Contents
4.6
(4-33)
Positive Integral Exponents
239
Note that the power 3 is applied to each factor of the product. This example illustrates the power of a product rule.
Power of a Product Rule
If a and b are real numbers and n is a positive integer, then
(ab)n a nb n.
E X A M P L E
3
Using the power of a product rule
Simplify. Assume that the variables are nonzero.
b) (3m)3
a) (xy3)5
c) (2x 3y 2z7)3
Solution
a) (xy 3)5 x 5(y 3)5 Power of a product rule
x 5y15
Power rule
3
3 3
b) (3m) (3) m Power of a product rule
27m3 (3)(3)(3) 27
c) (2x 3y 2z7)3 23(x 3)3(y 2)3(z7)3 8x9y6z 21
■
Power of a Quotient
helpful
Raising a quotient to a power is similar to raising a product to a power:
x 3 x x x
Definition of exponent 3
5
5 5 5
xxx
Definition of multiplication of fractions
555
x3
3
Definition of exponents
5
hint
Note that these rules of exponents are not absolutely
necessary. We could simplify
every expression here by
using only the definition of
exponent. However, these
rules make it a lot simpler.
The power is applied to both the numerator and denominator. This example illustrates the power of a quotient rule.
Power of a Quotient Rule
If a and b are real numbers, b 0, and n is a positive integer, then
a
b
E X A M P L E
4
Solution
2 2
22
a) 3 5x
(5x 3)2
4
6
25x
▲
▲
an
n.
b
Using the power of a quotient rule
Simplify. Assume that the variables are nonzero.
2 2
3x4 3
a) 3
b) 3
5x
2y
|
n
|
12a5b
c) 4a2b7
Power of a quotient rule
(5x 3)2 52(x 3)2 25x 6
e-Text Main Menu
|
Textbook Table of Contents
3
240
(4-34)
Chapter 4
Polynomials and Exponents
3x4
3
2y
33x12
23y9
27x12
8y 9
b)
3
Power of a quotient and power of a product rules
Simplify.
c) Use the quotient rule to simplify the expression inside the parentheses before
using the power of a quotient rule.
12a5b
4a2b7
3a3
b6
3
27a9
b18
3
Use the quotient rule first.
Power of a quotient rule
■
Summary of Rules
helpful
The rules for exponents are summarized in the following box.
hint
Note that the rules of exponents show how exponents
behave with respect to multiplication and division only. We
studied the more complicated
problem of using exponents
with addition and subtraction
in Section 4.4 when we
learned rules for (a b)2 and
(a b)2.
Rules for Nonnegative Integral Exponents
The following rules hold for nonzero real numbers a and b and nonnegative
integers m and n.
1. a0 1 Definition of zero exponent
2. am an amn Product rule
am
3. n amn for m n,
a
am
1
for n m
n a
anm
Quotient rule
4. (am)n amn Power rule
5. (ab)n an bn Power of a product rule
n
a
6. b
an
n
b
Power of a quotient rule
@ http://www.cne.gmu.edu/modules/dau/algebra/exponents/exp1_frm.html
WARM-UPS
True or false? Assume that all variables represent nonzero real
numbers. A statement involving variables is to be marked true only if it
is an identity. Explain your answer.
1. 30 1 False
4. (2x)4 2x4 False
7. (ab3)4 a4b12
2y3
10. 9
|
▲
▲
|
2
4y6
81
2. 25 28 413 False
5. (q3)5 q8 False
a12
True 8. 4 a3 False
a
3. 23 33 65 False
6. (3x 2)3 27x6 False
6w4
9. 9 2w5 False
3w
True
e-Text Main Menu
|
Textbook Table of Contents
4.6
4. 6
(4-35)
Positive Integral Exponents
241
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
Simplify. See Example 4.
2a
16a
37. b
2x y
39. 4y
6x y z
4
41. 3x y z
35.
1. What is the product rule for exponents?
9r
38. t
3y
40. 2zy
10rs t
42. 2rs t
3
x4
4
36.
2 4
2. What is the quotient rule for exponents?
3 2
3
n and
5
3
2
45. (3 6)2
49. (2 3)
For all exercises in this section, assume that the variables represent nonzero real numbers.
Simplify the exponential expressions. See Example 1.
51.
8. x6 x7 x
10. (3r4)(6r 2)
r
2a3
13. 4a7
2a 5b 3a7b3
15. 15a6b8
0
3t9
14. 6t18
18. 2 10
2
9
19. (x )
17. 23 52 200
x
20. (y )
2
5
2
1
58.
61. (5x4)3
62. (4z3)3
125x
5 6
27. (xy2)3 x
29. (2t5)3
28. (wy2)6 w
30. (3r3)3
31. (2x2y5)3
32. (3y2z3)3
(a4b2c5)3
33. a3b4c
(2ab2c3)5
34. 34
(2a bc)
▲
|
|
e-Text Main Menu
64z
64. 2a4b5 2a9b2
20a5b13
66. 4
5a b13
2x
x4
4 7
27y
3
16x
4
68. (ab)3(3ba2)4
9y
10a c
72. 5a b
5x yz
74. 5x yz
70.
3y8
y5
2
5
5
Simplify. See Example 3.
▲
7
3 4 5
5y3(y5)2
26. 5
10y (y2)6
1
2
60. 2y3(3y)
8a b
71. 4c 8x y
73. 16x y (r4)2
24. 53
(r )
3
3
3
Simplify each expression.
59. 3x4 5x7 15x
69.
22. (y2)6 3y5
2x
(t 2)5
23. 34
(t )
3x(x5)2
25. 6x3(x 2)4
56. 102 104 1,000,000
2 3
y
3
55. 23 24 128
67. (xt2)(2x2t)4
2 4
4
343
3
54. 103 33 27,000
9u4v9
65. 3u5v8
3
21. 2x 2 (x2)5
52.
63. 3y z 9yz
Simplify. See Example 2.
2 3
3
5 12
3xy 5xy
16. 20x3y14
8
2
50. (3 4)3
53. 52 23 200
57.
12. x y x y (x y)
3 6
5
48. 33 43 91
1
3
3 2
11. a3b4 ab6(ab)0 a
2
46. 52 32 16
81
47. 23 33 19
The power of a quotient rule says that
6
2 7
Simplify each expression. Your answer should be an integer or a
fraction. Do not use a calculator.
43. 32 62 45
44. (5 3)2 4
6. What is the power of a quotient rule?
9. (3u8)(2u2)
9 4 3
6 4 3
5. What is the power of a product rule?
The power of a product rule says that (
7. 22 25 128
2
2 4 9 2
4. What is the power rule for exponents?
4
8
2
3. Why must the bases be the same in these rules?
These rules do not make sense without identical bases.
3
y2
2
5
5 4
5
2
3
2
32c
b
3 5
z
Solve each problem.
75. Long-term investing. Sheila invested P dollars at annual
rate r for 10 years. At the end of 10 years her investment
was worth P(1 r)10 dollars. She then reinvested this
money for another 5 years at annual rate r. At the end
of the second time period her investment was worth
|
Textbook Table of Contents
242
(4-36)
Chapter 4
Polynomials and Exponents
P(1 r)10 (1 r)5 dollars. Which law of exponents can be
used to simplify the last expression? Simplify it.
P(1 r)
76. CD rollover. Ronnie invested P dollars in a 2-year CD
with an annual rate of return of r. After the CD rolled over
two times, its value was P((1 r)2)3. Which law of exponents can be used to simplify the expression? Simplify it.
P(1 r)
4.7
GET TING MORE INVOLVED
77. Writing. When we square a product, we square each factor
in the product. For example, (3b)2 9b2. Explain why we
cannot square a sum by simply squaring each term of the
sum.
78. Writing. Explain why we define 20 to be 1. Explain why
20 1.
NEGATIVE EXPONENTS AND
SCIENTIFIC NOTATION
We defined exponential expressions with positive integral exponents in Chapter 1
and learned the rules for positive integral exponents in Section 4.6. In this section
you will first study negative exponents and then see how positive and negative integral exponents are used in scientific notation.
In this
section
●
Negative Integral Exponents
●
Rules for Integral Exponents
Negative Integral Exponents
●
Converting from Scientific
Notation
●
Converting to Scientific
Notation
If x is nonzero, the reciprocal of x is written as 1. For example, the reciprocal of 23 is
x
written as 13. To write the reciprocal of an exponential expression in a simpler way,
●
Computations with
Scientific Notation
2
1
we use a negative exponent. So 23 23. In general we have the following definition.
Negative Integral Exponents
If a is a nonzero real number and n is a positive integer, then
1
an .
(If n is positive, n is negative.)
an
E X A M P L E
1
Simplifying expressions with negative exponents
Simplify.
a) 25
b) (2)5
2 3
c) 2
3
Solution
calculator
1
1
a) 25 5 2
32
1
b) (2)5 5 Definition of negative exponent
(2)
1
1
32
32
close-up
You can evaluate expressions
with negative exponents on a
calculator as shown here.
|
▲
▲
23
c) 2 23 32
3
1
1
3 2
2
3
1 1 1 9 9
8 9 8 1 8
|
e-Text Main Menu
|
■
Textbook Table of Contents
4.7
(4-37)
Negative Exponents and Scientific Notation
243
In simplifying 52, the negative sign preceding the 5 is used
after 5 is squared and the reciprocal is found. So 52 (52) 215.
CAUTION
To evaluate an, you can first find the nth power of a and then find the reciprocal. However, the result is the same if you first find the reciprocal of a and then find
the nth power of the reciprocal. For example,
1 2 1 1 1
1
1
or
32 .
32 2 3
9
3 3 9
3
helpful
So the power and the reciprocal can be found in either order. If the exponent is 1,
we simply find the reciprocal. For example,
hint
Just because the exponent
is negative, it doesn’t mean
the expression is negative.
1
51 ,
5
1
4,
3
5
1
and
5
.
3
Because 32 32 1, the reciprocal of 32 is 32, and we have
1
2
32.
3
1
Note that (2)3 while
(2)4 .
1
1
4
8
16
These examples illustrate the following rules.
Rules for Negative Exponents
If a is a nonzero real number and n is a positive integer, then
1 n
an ,
a
1
a1 ,
a
E X A M P L E
2
1
n
an, and
a
a
b
3
b) 101 101
Solution
3 3
4
a) 4
3
2
c) 103
3
64
27
1
1
2
1
b) 101 101 10 10 10 5
2
1
c) 2 2 103 2 1000 2000
103
103
calculator
close-up
You can use a calculator to
demonstrate that the product
rule for exponents holds when
the exponents are negative
numbers.
■
Rules for Integral Exponents
Negative exponents are used to make expressions involving reciprocals simpler
looking and easier to write. Negative exponents have the added benefit of working
in conjunction with all of the rules of exponents that you learned in Section 4.6. For
example, we can use the product rule to get
x2 x3 x2(3) x5
and the quotient rule to get
y3
5 y35 y2.
y
▲
▲
b n
.
a
Using the rules for negative exponents
Simplify.
3
a) 4
|
n
|
e-Text Main Menu
|
Textbook Table of Contents
244
(4-38)
Chapter 4
Polynomials and Exponents
With negative exponents there is no need to state the quotient rule in two parts as we
did in Section 4.6. It can be stated simply as
am
n amn
a
for any integers m and n. We list the rules of exponents here for easy reference.
helpful
Rules for Integral Exponents
hint
The following rules hold for nonzero real numbers a and b and any integers
m and n.
Definition of zero exponent
1. a0 1
m
n
mn
Product rule
2. a a a
m
a
3. n amn
Quotient rule
a
Power rule
4. (am)n amn
The definitions of the different
types of exponents are a really
clever mathematical invention. The fact that we have
rules for performing arithmetic with those exponents
makes the notation of exponents even more amazing.
5. (ab)n an bn
n
a
an
6. n
b
b
E X A M P L E
3
Power of a product rule
Power of a quotient rule
The product and quotient rules for integral exponents
Simplify. Write your answers without negative exponents. Assume that the variables represent nonzero real numbers.
a) b3b5
m6
c) 2
m
helpful
b) 3x3 5x 2
4y5
d) 12y3
Solution
a) b3b5 b35
b2
hint
Example 3(c) could be done
using the rules for negative
exponents and the old quotient rule:
1
m6 m2
m2 m6 m4
It is always good to look at
alternative methods.The more
tools in your toolbox the
better.
3
b) 3x
Product rule
Simplify.
5x 15x1
15
x
2
m6
c) 2 m6(2)
m
m4
1
4
m
Product rule
Definition of negative exponent
Quotient rule
Simplify.
Definition of negative exponent
4y5
y 5(3) y8
d) 12y3
3
3
■
|
▲
▲
In the next example we use the power rules with negative exponents.
|
e-Text Main Menu
|
Textbook Table of Contents
4.7
E X A M P L E
4
(4-39)
245
The power rules for integral exponents
Simplify each expression. Write your answers with positive exponents only. Assume that all variables represent nonzero real numbers.
4x5 2
b) (10x3)2
c) a) (a3)2
y2
Solution
a) (a3)2 a3 2 Power rule
a6
1
6
Definition of negative exponent
a
b) (10x3)2 102(x3)2 Power of a product rule
102x(3)(2) Power rule
x6
2
Definition of negative exponent
10
x6
100
5 2
(4x5)2
4x
c) Power of a quotient rule
2
(y 2)2
y
calculator
close-up
You can use a calculator to
demonstrate that the power
rule for exponents holds when
the exponents are negative
integers.
helpful
Negative Exponents and Scientific Notation
hint
The exponent rules in this section apply to expressions that
involve only multiplication
and division. This is not too
surprising since exponents,
multiplication, and division
are closely related. Recall that
a3 a a a and a b a b1.
42x10
y 4
Power of a product rule and power rule
1
42 x10 4
y
1
2 x10 y4
4
x10y4
16
a
1
Because a .
b
b
Definition of negative exponent
Simplify.
■
Converting from Scientific Notation
Many of the numbers occurring in science are either very large or very small. The
speed of light is 983,569,000 feet per second. One millimeter is equal to 0.000001
kilometer. In scientific notation, numbers larger than 10 or smaller than 1 are written by using positive or negative exponents.
Scientific notation is based on multiplication by integral powers of 10. Multiplying a number by a positive power of 10 moves the decimal point to the right:
10(5.32) 53.2
102(5.32) 100(5.32) 532
103(5.32) 1000(5.32) 5320
|
▲
▲
Multiplying by a negative power of 10 moves the decimal point to the left:
1
101(5.32) (5.32) 0.532
10
1
102(5.32) (5.32) 0.0532
100
1
3
10 (5.32) (5.32) 0.00532
1000
|
e-Text Main Menu
|
Textbook Table of Contents
246
(4-40)
Chapter 4
Polynomials and Exponents
So if n is a positive integer, multiplying by 10n moves the decimal point n places to
the right and multiplying by 10n moves it n places to the left.
A number in scientific notation is written as a product of a number between
1 and 10 and a power of 10. The times symbol indicates multiplication. For
example, 3.27 109 and 2.5 104 are numbers in scientific notation. In scientific notation there is one digit to the left of the decimal point.
To convert 3.27 109 to standard notation, move the decimal point nine places
to the right:
3.27 109 3,270,000,000
calculator
close-up
On a graphing calculator you
can write scientific notation
by actually using the power of
10 or press EE to get the letter
E, which indicates that the following number is the power
of 10.
9 places to the right
Of course, it is not necessary to put the decimal point in when writing a whole number.
To convert 2.5 104 to standard notation, the decimal point is moved four
places to the left:
2.5 104 0.00025
4 places to the left
In general, we use the following strategy to convert from scientific notation to
standard notation.
Note that if the exponent is
not too large, scientific notation is converted to standard
notation when you press
ENTER.
1. Determine the number of places to move the decimal point by examining
the exponent on the 10.
2. Move to the right for a positive exponent and to the left for a negative
exponent.
5
Converting scientific notation to standard notation
Write in standard notation.
b) 8.13 105
a) 7.02 106
Solution
a) Because the exponent is positive, move the decimal point six places to the right:
7.02 106 7020000. 7,020,000
b) Because the exponent is negative, move the decimal point five places to the left.
8.13 105 0.0000813
■
Converting to Scientific Notation
To convert a positive number to scientific notation, we just reverse the strategy for
converting from scientific notation.
Strategy for Converting to Scientific Notation
1. Count the number of places (n) that the decimal must be moved so that it
will follow the first nonzero digit of the number.
2. If the original number was larger than 10, use 10n.
3. If the original number was smaller than 1, use 10n.
|
▲
▲
E X A M P L E
Strategy for Converting from Scientific
Notation to Standard Notation
|
e-Text Main Menu
|
Textbook Table of Contents
4.7
Negative Exponents and Scientific Notation
(4-41)
247
Remember that the scientific notation for a number larger than 10 will have a
positive power of 10 and the scientific notation for a number between 0 and 1 will
have a negative power of 10.
E X A M P L E
6
Converting numbers to scientific notation
Write in scientific notation.
a) 7,346,200
b) 0.0000348
c) 135 1012
Solution
a) Because 7,346,200 is larger than 10, the exponent on the 10 will be positive:
calculator
7,346,200 7.3462 106
close-up
b) Because 0.0000348 is smaller than 1, the exponent on the 10 will be negative:
0.0000348 3.48 105
To convert to scientific notation, set the mode to scientific.
In scientific mode all results
are given in scientific notation.
c) There should be only one nonzero digit to the left of the decimal point:
135 1012 1.35 102 1012
1.35 1010
Convert 135 to scientific notation.
Product rule
■
Computations with Scientific Notation
An important feature of scientific notation is its use in computations. Numbers in
scientific notation are nothing more than exponential expressions, and you have already studied operations with exponential expressions in this section. We use the
same rules of exponents on numbers in scientific notation that we use on any other
exponential expressions.
E X A M P L E
7
Using the rules of exponents with scientific notation
Perform the indicated computations. Write the answers in scientific notation.
a) (3 106)(2 108)
4 105
b) 8 102
c) (5 107)3
calculator
close-up
With a calculator’s built-in scientific notation, some parentheses can be omitted as
shown below. Writing out the
powers of 10 can lead to errors.
Solution
a) (3 106)(2 108) 3 2 106 108 6 1014
4 105
4 105
1
5(2)
b) Quotient rule
2
2 10
8 10
8 10
2
(0.5)107
5 101 107 Write 0.5 in scientific notation.
5 106
Product rule
7 3
3
7 3
c) (5 10 ) 5 (10 )
Power of a product rule
21
125 10
Power rule
2
21
1.25 10 10
125 1.25 102
19
1.25 10
Product rule
▲
▲
Try these computations with
your calculator.
|
1
0.5
2
|
e-Text Main Menu
|
Textbook Table of Contents
■
248
(4-42)
Chapter 4
E X A M P L E
Polynomials and Exponents
8
Converting to scientific notation for computations
Perform these computations by first converting each number into scientific notation.
Give your answer in scientific notation.
a) (3,000,000)(0.0002)
b) (20,000,000)3(0.0000003)
Solution
a) (3,000,000)(0.0002) 3 106 2 104 Scientific notation
6 102
Product rule
3
7 3
b) (20,000,000) (0.0000003) (2 10 ) (3 107) Scientific notation
8 1021 3 107 Power of a product rule
24 1014
2.4 101 1014
24 2.4 101
2.4 1015
Product rule
WARM-UPS
■
True or false? Explain your answer.
1 1
2. 5 False
5
32 1
4. 1 True
3
3
6. 0.000036 3.6 105 True
8. 0.442 103 4.42 104 True
1
1. 102 True
100
3. 32 21 63
5.
7.
9.
10.
False
23.7 2.37 101 False
25 107 2.5 108 True
(3 109)2 9 1018 True
(2 105)(4 104) 8 1020
False
@ http://www.napanet.net/education/redwood/departments/mathhandbook/Default.htm#exponents
4.7
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What does a negative exponent mean?
A
2. What is the correct order for evaluating the operations indicated by a negative exponent?
The operations can be evaluated in any order.
3. What is the new quotient rule for exponents?
m
|
▲
▲
4. How do you convert a number from scientific notation to
standard notation?
Convert from scientific notation by multiplying by the appropriate power of 10.
5. How do you convert a number from standard notation to
scientific notation?
Convert from standard notation by counting the number of
places the decimal must move so that there is one nonzero
digit to the left of the decimal point.
|
e-Text Main Menu
6. Which numbers are not usually written in scientific notation?
Numbers between 1 and 10 are not written in scientific notation.
Variables in all exercises represent positive real numbers. Evaluate each expression. See Example 1.
7. 31
8. 33
10. (3)4
52
13. 102
11. 42
4
9. (2)4
12. 24
34
14. 2
6
Simplify. See Example 2.
5 3
4 2 9
16. 15. 17. 61 61
16
2
3
1
0
1
18. 21 41
19. 3 1250 20. 25 104
5
1
32
23
2
1
22. 702
21. 3
4
2
102 72
|
Textbook Table of Contents
1
4.7
Negative Exponents and Scientific Notation
(4-43)
249
Simplify. Write answers without negative exponents. See
Example 3.
23. x1x 2 x
24. y3y5
68. 5,670,000,000
69. 525 109
70. 0.0034 108
25. 2x2 8x6
26. 5y5(6y7)
Perform the computations. Write answers in scientific notation.
See Example 7.
27. 3a2(2a3)
28. (b3)(b5)
u5 1
29. u3 u8
8t3
31. 2t5
6x 5
33. 2
w4 1
30. w6 w10
22w4
32. 11w3
51y6
34. 17y9
71. (3 105)(2 1015)
72. (2 109)(4 1023)
4 108
73. 2 1030
9 104
74. 3 106
3 1020
75. 6 108
Simplify each expression. Write answers without negative exponents. See Example 4.
35. (x2)5
36. (y2)4
37. (a3)3
38. (b5)2 b
39. (2x3)4
40. (3y1)2
41. (4x2y3)2
2x1
43. y 3
1 108
76. 4 107
77. (3 1012)2
78. (2 105)3
79. (5 104)3
80. (5 1014)1
81. (4 1032)1
82. (6 1011)2 3.6
42. (6s2t4)1
2
a2
44. 3
3b
2a
45. ac
3
3w
46. wx
3 4
Perform the following computations by first converting each
number into scientific notation. Write answers in scientific notation. See Example 8.
83. (4300)(2,000,000)
2 2
2
4 3
Simplify. Write answers without negative exponents.
47. 21 31
48. 21 31
49. (2 31)1
50. (21 3)1
84.
85.
86.
87.
(40,000)(4,000,000,000)
(4,200,000)(0.00005) 2.1
(0.00075)(4,000,000)
(300)3(0.000001)5
51. (x2)3 3x7(5x1)
88. (200)4(0.0005)3
52. (ab1)2 ab(ab3)
(4000)(90,000)
89. 3
0.00000012
(30,000)(80,000)
90. 2
(0.000006)(0.002)
a3b2
b6a2
53. 1
b5
a
54.
x3y1
2x
3
2
6x9y3
3x3
Write each number in standard notation. See Example 5.
55. 9.86 109 9,860,000,000 56. 4.007 104 40,070
3
91.
92.
93.
94.
5
57. 1.37 10
0.00137
6
59. 1 10
0.000001
61. 6 105 600,000
58. 9.3 10
0.000093
1
60. 3 10
0.3
62. 8 106 8,000,000
|
▲
▲
Write each number in scientific notation. See Example 6.
63. 9000 9
64. 5,298,000
65. 0.00078 7.8
66. 0.000214
67. 0.0000085
|
e-Text Main Menu
95.
2
Perform the following computations with the aid of a
calculator. Write answers in scientific notation. Round
to three decimal places.
(6.3 106)(1.45 104)
(8.35 109)(4.5 103)
(5.36 104) (3.55 105)
(8.79 108) (6.48 109)
(3.5 105)(4.3 106)
3.4 108
(3.5 108)(4.4 104)
96. 2.43 1045
97. (3.56 1085)(4.43 1096)
98. (8 1099) (3 1099)
|
Textbook Table of Contents
(4-44)
Chapter 4
Polynomials and Exponents
Solve each problem.
99. Distance to the sun. The distance from the earth to the
sun is 93 million miles. Express this distance in feet.
(1 mile 5280 feet.) 4.910 10 feet
93 million miles
Sun
Earth
FIGURE FOR EXERCISE 99
100. Speed of light. The speed of light is 9.83569 108
feet per second. How long does it take light to travel from
the sun to the earth? See Exercise 99. 8.3 minutes
101. Warp drive, Scotty. How long does it take a spacecraft
traveling at 2 1035 miles per hour (warp factor 4) to
travel 93 million miles. 4.65 10 hours
102. Area of a dot. If the radius of a very small circle is
2.35 108 centimeters, then what is the circle’s area?
103. Circumference of a circle. If the circumference of a circle is 5.68 109 feet, then what is its radius?
10 4
10 3
Ag
102
V
101
Hg
1 Pb
Cr Ba
101
102
102
100
1
104
Concentration in waste (percent)
FIGURE FOR EXERCISE 106
107. Present value. The present value P that will amount
to A dollars in n years with interest compounded annually at annual interest rate r, is given by
P A (1 r )n.
Find the present value that will amount to $50,000 in
20 years at 8% compounded annually. $10,727.41
108. Investing in stocks. U.S. small company stocks have returned an average of 14.9% annually for the last 50 years
(T. Rowe Price, www.troweprice.com). Use the present
value formula from the previous exercise to find the
amount invested today in small company stocks that
would be worth $1 million in 50 years, assuming that
small company stocks continue to return 14.9% annually
for the next 50 years. $963.83
Value (millions of dollars)
108
Ra
106
104
Au
102
1
Cu
U
102
103 106 109
100 1
Concentration in ore (percent)
FIGURE FOR EXERCISE 105
|
▲
▲
Metal price (dollars/pound)
104. Diameter of a circle. If the diameter of a circle is
1.3 1012 meters, then what is its radius?
6.5 10 meters
105. Extracting metals from ore. Thomas Sherwood studied
the relationship between the concentration of a metal in
commercial ore and the price of the metal. The accompanying graph shows the Sherwood plot with the locations of
several metals marked. Even though the scales on this
graph are not typical, the graph can be read in the same
manner as other graphs. Note also that a concentration of
100 is 100%.
a) Use the figure to estimate the price of copper (Cu) and
its concentration in commercial ore.
$1 per pound and 1%
b) Use the figure to estimate the price of a metal that has
a concentration of 106 percent in commercial ore.
$1,000,000 per pound.
c) Would the four points shown in the graph lie along a
straight line if they were plotted in our usual coordinate system? No
106. Recycling metals. The accompanying graph shows the
prices of various metals that are being recycled and the
minimum concentration in waste required for recycling.
The straight line is the line from the figure for Exercise 105. Points above the line correspond to metals for
which it is economically feasible to increase recycling
efforts.
a) Use the figure to estimate the price of mercury (Hg)
and the minimum concentration in waste required for
recycling mercury.
$1 per pound and 1%
b) Use the figure to estimate the price of silver (Ag) and
the minimum concentration in waste required for recycling silver.
$100 per pound and 0.1%
Metal price (dollars/pound)
250
|
e-Text Main Menu
1
Amount after
50 years, $1 million
0.5
Present value, P
0
0
10
20 30
Years
40 50
FIGURE FOR EXERCISE 108
|
Textbook Table of Contents
Chapter 4
GET TING MORE INVOLVED
Collaborative Activities
(4-45)
251
110. Discussion. Which of the following expressions is not
equal to 1? Explain your answer.
a) 11
b) 12
c) (11)1
d) (1)1
2
e) (1)
e
3
109. Exploration. a) If w 0, then what can you say about
w? b) If (5)m 0, then what can you say about m?
c) What restriction must be placed on w and m so that
w m 0?
a) w 0 b) m is odd c) w 0 and m odd
COLLABORATIVE ACTIVITIES
Area as a Model of FOIL
Grouping: Pairs
Topic: Multiplying polynomials
Sometimes we can use drawings to represent mathematical operations. The area of a rectangle can represent the process we
use when multiplying binomials. The rectangle below represents the multiplication of the binomials (x 3) and (x 5):
3x
x
2
5x
x
5
(x + 3)
x
3.
The areas of the inner rectangles are x2, 3x, 5x, and
15.
15
3
For problem 3, student B uses FOIL and A uses the diagram.
5
The area of the red rectangle equals the sum of the
areas of the four inner rectangles.
(x + 5)
(x 5)(x 4) ?
(x + 5)
x
x
4
(x + 4)
Area of red rectangle:
(x 3)(x 5) x 3x 5x 15
2
4. StudentAdraws a diagram to find the product (x 3)(x 7).
Student B finds (x 3)(x 7) using FOIL.
x 2 8x 15
1. a. With your partner, find the areas of the inner rectangles to
find the product (x 2)(x 7) below:
5. Student B draws a diagram to find the product (x 2)(x 1).
Student A finds (x 2)(x 1) using FOIL.
Thinking in reverse: Work together to complete the product
that is represented by the given diagram.
2
(x + 2)
x
6.
x
9
7
(x + ?)
(x + 7)
x2
(x 2)(x 7) ?
b. Find the same product (x 2)(x 7) using FOIL.
(x + ?)
For problem 2, student A uses FOIL to find the given product while student B finds the area with the diagram.
7.
10
2.
(x + ?)
x2
8
(x 8)(x 1) ?
(x + 8)
(x + ?)
x
x
Extension: Make up a FOIL problem, then have your partner
draw a diagram of it.
1
|
▲
▲
(x + 1)
|
e-Text Main Menu
|
Textbook Table of Contents
252
(4-46)
Chapter 4
Polynomials and Exponents
C H A P T E R
W R A P - U P
4
SUMMARY
Polynomials
Examples
Term
A number or the product of a number and
one or more variables raised to powers
5x3, 4x, 7
Polynomial
A single term or a finite sum of terms
2x 5 9x 2 11
Degree of a polynomial The highest degree of any of the terms
Degree of 2x 9 is 1.
Degree of 5x 3 x 2 is 3.
Adding, Subtracting, and Multiplying Polynomials
Examples
Add or subtract
polynomials
Add or subtract the like terms.
(x 1) (x 4) 2x 3
(x2 3x) (4x2 x)
3x2 2x
Multiply monomials
Use the product rule for exponents.
2x5 6x8 12x13
Multiply polynomials
Multiply each term of one polynomial by
every term of the other polynomial, then
combine like terms.
x 2 2x 5
x1
2
x 2x 5
x 3 2x 2 5x
x 3 x 2 3x 5
Binomials
Examples
FOIL
A method for multiplying two binomials quickly
(x 2)(x 3) x2 x 6
Square of a sum
(a b)2 a2 2ab b2
(x 3)2 x2 6x 9
Square of a difference
(a b)2 a2 2ab b2
(m 5)2 m2 10m 25
Product of a sum
and a difference
(a b)(a b) a2 b2
(x 2)(x 2) x2 4
Dividing Polynomials
Examples
Dividing monomials
Use the quotient rule for exponents
8x5 (2x2) 4x3
Divide a polynomial
by a monomial
Divide each term of the polynomial by the
monomial.
3x5 9x
x4 3
3x
x 7 ← Quotient
If the divisor is a binomial, use
Divisor → x 2x2
x
5
4 ← Dividend
long division.
x2 2x
(divisor)(quotient) (remainder) dividend
7x 4
7x 14
10 ← Remainder
|
▲
▲
Divide a polynomial
by a binomial
|
e-Text Main Menu
|
Textbook Table of Contents
Chapter 4
Rules of Exponents
Summary
(4-47)
Examples
The following rules hold for any integers m and n,
and nonzero real numbers a and b.
Zero exponent
a0 1
20 1, (34)0 1
Product rule
am an amn
a2 a3 a5
3x6 4x9 12x15
Quotient rule
am
n amn
a
y3
x8 x2 x6, 3 y0 1
y
c7
1
9 c2 2
c
c
(am)n amn
Power rule
(22)3 26, (w5)3 w15
Power of a product rule (ab)n anbn
a
b
Power of a
quotient rule
n
(2t)3 8t3
an
n
b
x
3
Negative Exponents
3
x3
27
Examples
Negative integral
exponents
If n is a positive integer and a is a nonzero real
1
number, then an n
a
Rules for negative
exponents
If a is a nonzero real number and n is a positive
n
1
1
1
integer, then an , a1 , and n
an.
a
a
a
Scientific Notation
1 5
1
5
32 ,
2 x
3
x
2 3
3 3
1
, 51 3
2
5
1
8
w8
w
Examples
Converting from
scientific notation
1. Find the number of places to move the decimal
point by examining the exponent on the 10.
2. Move to the right for a positive exponent and to
the left for a negative exponent.
1. Count the number of places (n) that the decimal
point must be moved so that it will follow the
first nonzero digit of the number.
2. If the original number was larger than 10,
use 10n.
3. If the original number was smaller than 1,
use 10n.
|
▲
▲
Converting into
scientific notation
(positive numbers)
|
e-Text Main Menu
|
5.6 103 5600
9 104 0.0009
304.6 3.046 102
0.0035 3.5 103
Textbook Table of Contents
253
254
(4-48)
Chapter 4
Polynomials and Exponents
ENRICHING YOUR MATHEMATICAL WORD POWER
1. term
a. an expression containing a number or the product of
a number and one or more variables
b. the amount of time spent in this course
c. a word that describes a number
d. a variable a
c. an equation with no solution
d. a polynomial with five terms
b
7. dividend
a. a in ab
b. b in ab
c. the result of ab
d. what a bank pays on deposits
a
2. polynomial
a. four or more terms
b. many numbers
c. a sum of four or more numbers
d. a single term or a finite sum of terms
8. divisor
a. a in ab
b. b in ab
c. the result of ab
d. two visors b
For each mathematical term, choose the correct meaning.
d
3. degree of a polynomial
a. the number of terms in a polynomial
b. the highest degree of any of the terms of a polynomial
c. the value of a polynomial when x 0
d. the largest coefficient of any of the terms of a polynomial b
4. leading coefficient
a. the first coefficient
b. the largest coefficient
c. the coefficient of the first term when a polynomial is
written with decreasing exponents
d. the most important coefficient c
5.
monomial
a. a single polynomial
b. one number
c. an equation that has only one solution
d. a polynomial that has one term d
9.
quotient
a. a in ab
b. b in ab
c. ab
d. the divisor plus the remainder
c
10. binomial
a. a polynomial with two terms
b. any two numbers
c. the two coordinates in an ordered pair
d. an equation with two variables a
11. integral exponent
a. an exponent that is an integer
b. a positive exponent
c. a rational exponent
d. a fractional exponent a
12. scientific notation
a. the notation of rational exponents
b. the notation of algebra
c. a notation for expressing large or small numbers with
powers of 10
d. radical notation c
6. FOIL
a. a method for adding polynomials
b. first, outer, inner, last
REVIEW EXERCISES
4.1 Perform the indicated operations.
1. (2w 6) (3w 4) 5w 2
2. (1 3y) (4y 6) y 5
3. (x2 2x 5) (x2 4x 9)
x
4. (3 5x x2) (x2 7x 8)
5. (5 3w w2) (w2 4w 9)
6. (2t2 3t 4) (t2 7t 2)
7. (4 3m m2) (m2 6m 5)
8. (n3 n2 9) (n4 n3 5) n
4
5
13. x 5(x 3)
x 15
15. 5x 3(x2 5x 4)
14. x 4(x 9)
3x 36
16. 5 4x2(x 5)
17. 3m2(5m3 m 2)
18. 4a4(a2 2a 4)
4
study
2
1
4
Note how the review exercises are arranged according to the
sections in this chapter. If you are having trouble with a certain
type of problem, refer back to the appropriate section for examples and explanations.
4.2 Perform the indicated operations.
10. 3h3t2 2h2t5
12. (12b3)2 144
|
▲
▲
9. 5x2 (10x9)
11. (11a7)2
|
e-Text Main Menu
tip
|
Textbook Table of Contents
Chapter 4
19. (x 5)(x2 2x 10) 20. (x 2)(x2 2x 4)
61. (x3 x2 11x 10) (x 1)
62. (y3 9y2 3y 6) (y 1)
21. (x2 2x 4)(3x 2) 22. (5x 3)(x2 5x 4)
27. (4y 3)(5y 2)
24. (w 5)(w 12)
w 60
26. (5r 1)(5r 2)
r 2
28. (11y 1)(y 2)
29. (3x 5)(2x 1)
30. (x 7)(2x 7)
25. (2t 3)(t 9)
2
2
3
2x
63. x3
6
x 3
2
2x
65. 1x
3
2
1 x
2
4.4 Perform the indicated operations. Try to write only the
x2 3
67. x1
answers.
31. (z 7)(z 7)
z
49
33. (y 7)2
y
14y 49
35. (w 3)2
w
6w 9
37. (x2 3)(x2 3)
32. (a 4)(a 4)
a
16
34. (a 5)2
a
10a 25
36. (a 6)2
a
12a 36
38. (2b2 1)(2b2 1)
39. (3a 1)
9a
6a
41. (4 y)2
16
y
40. (1 3c)
1 6c
42. (9 t)2
81 18t
2
x
x
x 3
6
72. (3a5)(5a3)
30k3y9
74. 15k3y2
76. (y5)8 y
78. (3a4b6)4
3y 3
80. 2
3a4b8 4
82. 3
6a b12
4.7 Simlify each expression. Use only positive exponents in
answers.
83. 25
4m
1
▲
|
e-Text Main Menu
84. 24
32
86. 51 50
8
1
a8
89. a
a 12
b2
92. 6 b
b
Find the quotient and remainder.
55. (3m3 9m2 18m) (3m)
m
56. (8x3 4x2 18x) (2x)
9, 0
57. (b2 3b 5) (b 2) b
58. (r2 5r 9) (r 3) r 2, 3
59. (4x2 9) (2x 1) 2x 1,
60. (9y3 2y) (3y 2)
2, 4
▲
x
1
For the following exercises, assume that all of the variables
represent positive real numbers.
1
(a 1) (1 a)
1
(t 3) (3 t)
1
(m4 16) (m 2) m
2m
(x4 1) (x 1) x
x
x
|
1
1
8x3y5 4x2y4 2xy3
50. 2xy2
51.
52.
53.
54.
19
remainder
quotient .
divisor
3x
64. x4
12
3 x 4
3x
66. 5x
15
5 x
x2 3x 1
68. x3
19
x 6 1
x 3
2x2
70. x3
71. 2y10 3y20 6
10b5c3
5
73. 5
2b c9
75. (b5)6 b
77. (2x3y2)3
8x
2a 3
79. b
6x2y5 3 8x
81. 3z6
44. 6x4y2 (2x2y2) 3x
9h5t9r2
t
46. 7
3h t6r2
h
7y
48. y 7
1
2x
2x 9, 1
10
4.6 Simplify each expression.
4.5 In Exercises 43–54, find each quotient.
43. 10x5 (2x3)
5x
6a5b7c6
a
45. 3
3a b9c6
b
3x 9
3
47. 3
9x3 6x2 3x
x
49. 3x
2
x
1
x2
69. x1
2
1
255
Write each expression in the form
4.3 Perform the indicated operations.
23. (q 6)(q 8)
(4-49)
Review Exercises
95. (2x3)3
x
1
1000
85. 103
87. x5x8
88. a3a9
a10
90. 4
a
a3
91. 7
a
93. (x3)4
94. (x5)10
96. (3y5)2
a
97. 3b3
2 3
a
98. 5b
|
1
Textbook Table of Contents
2
256
(4-50)
Chapter 4
Polynomials and Exponents
Convert each number in scientific notation to a number in standard notation, and convert each number in standard notation to
a number in scientific notation.
99. 5000
100. 0.00009
10
9 10
102. 5.7 108
101. 3.4 105
340,000
0.000000057
103. 0.0000461
104. 44,000
4.61 10
4.4 10
6
106. 5.5 109
105. 5.69 10
0.00000569
5,500,000,000
140. (2x2 7x 4)(x 3)
141. (x2 4x 12) (x 2)
142. (a2 3a 10) (a 5)
Solve each problem.
143. Roundball court. The length of a basketball court is
44 feet larger than its width. Find polynomials that represent its perimeter and area. The actual width of a basketball court is 50 feet. Evaluate these polynomials to find
the actual perimeter and area of the court.
Perform each computation without using a calculator. Write answers in scientific notation.
107. (3.5 108)(2.0 1012) 7 10
108. (9 1012)(2 1017) 1.8
109. (2 104)4 1.6
110. (3 105)3
111. (0.00000004)(2,000,000,000) 8
112. (3,000,000,000) (0.000002) 1.5 10
113. (0.0000002)5 3.2
114. (50,000,000,000)3 1.25
(5x2 8x 8) (4x2 x 3)
5
(4x2 6x 8) (9x2 5x 7)
(2x2 2x 3) (3x2 x 9) 5x
(x2 3x 1) (x2 2x 1) 2x
19x
(x 4)(x2 5x 1) x
|
▲
▲
135.
136.
137.
138.
139.
1000
|
Weekly revenue
(hundreds of dollars)
(k 5)(k 4)
k
k 20
(t 7z)(t 6z)
t
13tz 42z
(4y2 9)0
1
(9y3c4)2
81y c
(3x 5)(2x 6)
6x
8x 30
2
(9x 2)(9x2 2)
81x
4
(4v 3)2
16v
24v 9
(k 10)3
k
k
300k
5w3r2 2w4r8
10w
6h3y5 4
7
2h y2
81y
h
FIGURE FOR EXERCISE 143
144. Badminton court. The width of a badminton court is
24 feet less than its length. Find polynomials that represent its perimeter and area. The actual length of a badminton court is 44 feet. Evaluate these polynomials to
find the perimeter and area of the court.
P 4x 48, A x
24x, P 128 ft, A 880 ft
145. Smoke alert. A retailer of smoke alarms knows that at a
price of p dollars each, she can sell 600 15p smoke
alarms per week. Find a polynomial that represents
the weekly revenue for the smoke alarms. Find the revenue for a week in which the price is $12 per smoke
alarm. Use the bar graph to find the price per smoke alarm
that gives the maximum weekly revenue.
R
15p
600p, $5040, $20
MISCELL ANEOUS
Perform the indicated operations.
115. (x 3)(x 7)
116.
x
10x 21
117. (t 3y)(t 4y)
118.
t
7ty 12y
120.
119. (2x3)0 (2y)0
2
122.
121. (3ht6)3
27h
123. (2w 3)(w 6)
124.
2w
9w 18
125. (3u 5v)(3u 5v) 126.
9u
v
2
127. (3h 5)
128.
9h
25
130.
129. (x 3)3
x
9x
x 27
132.
131. (7s2t)(2s3t5)
14s
k4m2 4
134.
133. 22
2k m
w 44 ft
w ft
70
60
50
40
30
20
10
0
4
8
12 16 20 24
28 32 36
Price (dollars)
FIGURE FOR EXERCISE 145
146. Boom box sales. A retailer of boom boxes knows that at
a price of q dollars each, he can sell 900 3q boom
boxes per month. Find a polynomial that represents the
monthly revenue for the boom boxes? How many boom
boxes will he sell if the price is $300 each?
R
q
900q, 0
e-Text Main Menu
|
Textbook Table of Contents
Chapter 4
Test
(4-51)
257
CHAPTER 4 TEST
Perform the indicated operations.
1. (7x3 x2 6) (5x2 2x 5)
2. (x2 3x 5) (2x2 6x 7)
6y3 9y2
3. 3y
4. (x 2) (2 x)
1
5. (x3 2x2 4x 3) (x 3)
6. 3x2(5x3 7x2 4x 1) 15x
study
Before you take an in-class exam on this chapter, work the sample test given here. Set aside one hour to work this test and use
the answers in the back of this book to grade yourself. Even
though your instructor might not ask exactly the same questions, you will get a good idea of your test readiness.
Find the products.
7. (x 5)(x 2)
x
10
9. (a 7)2
a
14a 49
11. (b 3)(b 3)
b
9
13. (4x2 3)(x2 2)
4x
x
6
tip
8. (3a 7)(2a 5)
6a
35
10. (4x 3y)2
16x
24xy 9y
12. (3t2 7)(3t2 7)
9t
49
14. (x 2)(x 3)(x 4)
x
3
x 24
25. (3s3t2)2
26. (2x6y)3
Convert to scientific notation.
27. 5,433,000 5.433 10
28. 0.0000065
Perform each computation by converting to scientific notation.
Give answers in scientific notation.
29. (80,000)(0.000006)
30. (0.0000003)4
4.8 10
Write each expression in the form
Solve each problem.
remainder
quotient .
divisor
2x
x2 3x 5
16. 15. x3
x2
6
15
x
x 3
x 2
Use the rules of exponents to simplify each expression. Write
answers without negative exponents.
17. 5x3 7x5
x
18. 3x3y (2xy4
6 5
5
19. 4a b (2a b)
ab 20. 3x2 5x7
5
2a
2a
6a7b6c2
21. 22. 3
2
b
b
2a b8c2
7
6t
w6
23. 24. 4
9
2t
w
31. Find the quotient and remainder when x2 5x 9 is divided by x 3.
x 2, 3
32. Subtract 3x2 4x 9 from x2 3x 6.
x
x 15
33. The width of a pool table is x feet, and the length is 4 feet
longer than the width. Find polynomials that represent the
area and the perimeter of the pool table. Evaluate these
polynomials for a width of 4 feet.
x
x ft x 8 ft, 32 ft
34. If a manufacturer charges q dollars each for footballs, then
he can sell 3000 150q footballs per week. Find a polynomial that represents the revenue for one week. Find the
weekly revenue if the price is $8 for each football.
R
q
3000q, $14,400
|
▲
▲
|
e-Text Main Menu
|
Textbook Table of Contents
MAKING CONNECTIONS CHAPTERS 1–4
1.
3.
5.
7.
8.
9.
11.
16 (2)
2. (2)3 1
2
(5) 3(5) 1
4. 210 215
15
10
2 2
6. 210 25
32 42
(172 85) (85 172)
(5 3)2
10. 52 32
(30 1)(30 1)
12. (30 1)2
Perform the indicated operations.
13. (x 3)(x 5)
14. (x2 8x 15) (x 5)
15. x 3(x 5)
16. (x2 8x 15)(x 5)
17. 5t3v 3t2v6
18. (10t3v2) (2t2v)
19. (6y3 8y2) (2y2)
20. (y2 3y 9) (3y2 2y 6)
Solve each equation.
21. 2x 1 0
22. x 7 0
23. 2x 3 0
24. 3x 7 5
25. 8 3x x 20
Solve the problem.
27. Average cost. Pineapple Recording plans to spend
$100,000 to record a new CD by the Woozies and $2.25 per
CD to manufacture the disks. The polynomial 2.25n 100,000 represents the total cost in dollars for recording
and manufacturing n disks. Find an expression that represents the average cost per disk by dividing the total cost
by n. Find the average cost per disk for n 1000, 100,000,
and 1,000,000. What happens to the large initial investment
of $100,000 if the company sells one million CDs?
6
Average cost (dollars)
Simplify each expression.
4
3
2
1
0
26. 4 3(x 2) 0
3
5
0
0.5
1
Number of disks (millions)
FIGURE FOR EXERCISE 27
@
Net Tutor
(4-52)
|
▲
▲
258
|
e-Text Main Menu
|
Textbook Table of Contents