Download 4.5 ADDITION AND SUBTRACTION OF POLYNOMIALS

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Chapter 4
4-46
Exponents and Polynomials
4.5
ADDITION AND SUBTRACTION OF POLYNOMIALS
To Succeed, Review How To . . .
1. Add and subtract like terms
(pp. 98–101).
Objectives
A Add polynomials.
B Subtract polynomials.
2. Remove parentheses in expressions
preceded by a minus sign (p. 101).
C Find areas by adding polynomials.
D Solve applications.
GETTING STARTED
The annual amount of waste (in millions of tons) generated in the United States is approximated by G(t) 0.001t3 0.06t2 2.6t 88.6, where t is the number of years
after 1960. How much of this waste is recovered? That amount can be
Waste Generated
approximated by R(t) 0.06t2 0.59t 6.4. From these two approximations, we can estimate that the amount of waste actually “wasted” (not
recovered) is G(t) R(t). To find this difference, we simply subtract like
terms. To make the procedure more familiar, we write it in columns:
300
Tons (in millions)
Wasted Waste
200
G(t) 0.001t3 0.06t2 2.6t 88.6
() R(t) ()
100
0.001t3 3.19t 82.2
Thus the amount of waste generated and not recovered is
0
0
10
20
30
35
40
Year (1960 0)
G(t) R(t) 0.001t3 3.19t 82.2.
Let’s see what this means in millions of tons. Since t is the number of
years after 1960, t 0 in 1960, and the amount of waste generated, the amount
of waste recycled, and the amount of waste not recovered are as follows:
Waste Recovered
80
Tons (in millions)
0.06t2 0.59t 6.4
G(0) 0.001(0)3 0.06(0)2 2.6(0) 88.6 88.6
(million tons)
60
R(0) 0.06(0)2 0.59(0) 6.4 6.4
G(0) R(0) 88.6 6.4 82.2
40
20
0
0
10
20
30
35
Year (1960 0)
40
(million tons)
(million tons)
As you can see, there is much more material not recovered than material
recovered. How can we find out whether the situation is changing? One way
is to predict how much waste will be produced and how much recovered,
say, in the year 2010. The amount can be approximated by G(50) R(50).
Then we find out if, percentagewise, the situation is getting better. In 1960,
the percent of materials recovered was 6.488.6, or about 7.2%. What percent would it be in the year 2010? In this section we will learn how to add
and subtract polynomials and use these ideas to solve applications.
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4.5
A
Addition and Subtraction of Polynomials
337
Adding Polynomials
The addition of monomials and polynomials is a matter of combining like terms. For
example, suppose we wish to add 3x 2 7x 3 and 5x 2 2x 9; that is, we wish to find
(3x 2 7x 3) (5x 2 2x 9)
Web It
Using the commutative, associative, and distributive properties, we write
To practice combining like
terms, go to link 4-5-1 on the
Bello Website at mhhe.com/
bello.
(3x 2 7x 3) (5x 2 2x 9) (3x 2 5x 2) (7x 2x) (3 9)
(3 5)x 2 (7 2)x (3 9)
8x 2 5x 6
3
1
Similarly, the sum of 4x 3 7 x 2 2x 3 and 6x 3 7 x 2 9 is written as
4x
3
3
1
x 2 2x 3 6x 3 x 2 9
7
7
3
1
(4x 3 6x 3) x 2 x 2 (2x) (3 9)
7
7
2
10x 3 x 2 2x 12
7
In both examples, the polynomials have been written in descending order for convenience
in combining like terms.
EXAMPLE 1
PROBLEM 1
Adding polynomials
Add: 3x 7x 2 7 and 4x 2 9 3x
Add: 5x 8x 2 3 and 3x 2 8 5x
SOLUTION We first write both polynomials in descending order and then
combine like terms to obtain
(7x 2 3x 7) (4x 2 3x 9) (7x 2 4x 2) (3x 3x) (7 9)
3x 2 0 2
3x 2 2
As in arithmetic, the addition of polynomials can be done by writing the polynomials
in descending order and then placing like terms in columns. In arithmetic, you add 345 and
678 by writing the numbers in a column:
345
678
Units
Tens
Hundreds
Thus to add 4x 3 3x 7 and 7x 3x 3 x 2 9, we first write both polynomials in
descending order with like terms in the same column, leaving space for any missing terms.
We then add the terms in each of the columns:
The x 2 term is missing in 4x 3 3x 7.
Answer
1. 5x 2 5
4x 3
3x 7
3x 3 x 2 7x 9
x 3 x 2 10x 2
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Chapter 4
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Exponents and Polynomials
EXAMPLE 2
PROBLEM 2
More practice adding polynomials
Add: 3x 7x 2 2 and 4x 2 3 5x
Add: 5y 8y 2 3 and
5y 2 4 6y
SOLUTION We first write both polynomials in descending order, place like
terms in a column, and then add as shown:
7x 2 3x 2
4x 2 5x 3
3x 2 2x 5
Horizontally, we write:
(7x 2 3x 2) (4x 2 5x 3)
14243
(7x 2 4x 2)
14243
(3x 5x) 1
(2
3)
42
43
3x 2
3x 2
B
2x
2x
(5)
5
Subtracting Polynomials
To subtract polynomials, we first recall that
a (b c) a b c
Web It
To practice with the subtraction of polynomials, go to link
4-5-2 on the Bello Website at
mhhe.com/bello and enter
your problem. Be careful with
parentheses!
For a more conventional approach and different strategies, try link 4-5-3.
To remove the parentheses from an expression preceded by a minus sign, we must change
the sign of each term inside the parentheses. This is the same as multiplying each term
inside the parentheses by 1. Thus
(3x 2 2x 1) (4x 2 5x 2) 3x 2 2x 1 4x 2 5x 2
(3x 2 4x 2) (2x 5x) (1 2)
x 2 7x (1)
x 2 7x 1
Here’s how we do it using columns:
3x 2 2x 1
3x 2 2x 1
is written
() 4x 2 5x 2
() 4x 2 5x 2
x 2 7x 1
Note that we changed
the sign of every term
in 4x 2 5x 2 and
wrote 4x 2 5x 2.
NOTE
“Subtract b from a” means to find a b.
EXAMPLE 3
PROBLEM 3
Subtracting polynomials
Subtract 4x 3 7x 2 from 5x 2 3x.
SOLUTION
and add:
Subtract 5y 4 8y 2 from 6y 2 4y.
We first write the problem in columns, then change the signs
5x 2 3x
is written
()7x 2 4x 3
5x 2 3x
()7x 2 4x 3
2x 2 7x 3
Thus the answer is 2x 2 7x 3.
Answers
2. 3y 2 y 7
3. 2y 2 9y 4
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Addition and Subtraction of Polynomials
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To do it horizontally, we write
(5x 2 3x) (7x 2 4x 3)
5x 2 3x 7x 2 4x 3
Change the sign of every term in
7x 2 4x 3.
(5x 2 7x 2) (3x 4x) 3
Use the commutative and associative properties.
2x 2 7x 3
Just as in arithmetic, we can add or subtract more than two polynomials. For example,
to add the polynomials 7x x 2 3, 6x 2 8 2x, and 3x x 2 5, we simply write
each of the polynomials in descending order with like terms in the same column and add:
x 2 7x 3
6x 2 2x 8
x 2 3x 5
6x 2 2x 6
Or, horizontally, we write
(x 2 7x 3) (6x 2 2x 8) (x 2 3x 5)
(x 2 6x 2 x 2) (7x 2x 3x) (3 8 5)
6x 2 (2x) (6)
6x 2 2x 6
EXAMPLE 4
PROBLEM 4
More practice adding polynomials
Add: x 2x 3x 5, 8 2x 5x , and 7x 4x 9
3
2
2
Add: y 3 3y 4y 2 6,
9 3y 6y 2, and
6y 3 5y 8
3
SOLUTION We first write all the polynomials in descending order with like
terms in the same column and then add:
x 3 3x 2 2x 5
5x 2 2x 8
3
7x
4x 9
8x 3 8x 2
4
Horizontally, we have
(x 3 3x 2 2x 5) (5x 2 2x 8) (7x 3 4x 9)
(x 3 7x 3) (3x 2 5x 2) (2x 2x 4x) (5 8 9)
8x 3 (8x 2) 0x (4)
8x 3 8x 2 4
C
Finding Areas
Addition of polynomials can be used to find the sum of the areas of several rectangles. To
find the total area of the shaded rectangles, add the individual areas.
5
2
Answer
4. 7y 10y y 7
3
2
A
5
3
B
2
2
D
C
3
5
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Chapter 4
4-50
Exponents and Polynomials
Web It
Since the area of a rectangle is the product of its length and its width, we have
To practice adding polynomials using sums of areas, go
back to link 4-5-3 on the
Bello Website at mhhe.com/
bello.
Area of A 14243
Area of B Area
of C Area
of D
14243
14243
14243
52 23 32 55
10
6
6
25
Thus
10 6 6 25 47
(square units)
This same procedure can be used when some of the lengths are represented by variables, as shown in the next example.
EXAMPLE 5
PROBLEM 5
Finding sums of areas
Find the sum of the areas of the shaded rectangles:
3 B
x
SOLUTION
3x
x
A
5
Find the sum of the areas of the shaded
rectangles:
x
C
3
D
3 B
y
A
3x
y
6
The total area in square units is
Area
of A 14243
Area of B 14243
Area of C 14243
Area of D
14243
5x
3x
3x
1444442444443
11x
(3x)2
9x 2
4y
y
D
C
3
4y
or
9x 2 11x
D
In descending order
Solving Applications
The concepts we’ve just studied can be used to examine important issues in American life.
Let’s see how we can use these concepts to study health care.
EXAMPLE 6
PROBLEM 6
Paying to get well
Based on Social Security Administration statistics, the annual amount of
money spent on hospital care by the average American can be approximated by
H(t) 6.6t2 30t 682
How much was spent on hospital care
and doctors’ services in 2000?
(dollars)
while the annual amount spent for doctors’ services is approximated by
D(t) 41t 293
where t is the number of years after 1985.
a. Find the annual amount of money spent for doctors’ services and hospital
care.
b. How much was spent for doctors’ services in 1985?
c. How much was spent on hospital care in 1985?
d. How much would you predict will be spent on hospital care and doctors’
services in the year 2005?
Answers
5. 12y 16y 2
6. $3525
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SOLUTION
a. To find the annual amount spent for doctors’ services and hospital care, we
find D(t) H(t):
D(t) 41t 293
H(t) 6.6t 2 30t 682
6.6t 2 71t 975
Thus the annual amount spent for doctors’ services and hospital care is
6.6t2 71t 975.
D(t)
H(t)
b. To find how much was spent for doctors’ services in 1985 (t 0), find
D(0) 41(0) 293. Thus $293 was spent for doctors’ services in 1985.
c. The amount spent on hospital care in 1985 (t 0) was H(0) 6.6(0)2 30(0) 682 682. Thus $682 was spent for hospital care in 1985.
d. The year 2005 corresponds to t 2005 1985 20. To predict the
amount to be spent on hospital care and doctors’ services in the year 2005,
we find 6.6(20)2 71(20) 975 $5035. Note that it’s easier to
evaluate D(t) H(t) 6.6t2 71t 975 for t 20 than it is to evaluate
D(20) and H(20) and then add.
EXAMPLE 7
More drinking
In Example 7 of Section 4.4, we introduced the polynomial equations
y 0.0226x 0.1509 and y 0.0257x 0.1663 approximating
the blood alcohol level (BAL) for a 150-pound male or female, respectively.
In these equations x represents the time since consuming 3 ounces of alcohol.
It is known that the burn-off rate of alcohol is 0.015 per hour (that is, the
BAL is reduced by 0.015 per hour if no additional alcohol is consumed). Find
a polynomial that would approximate the BAL for a male x hours after
consuming the 3 ounces of alcohol.
SOLUTION The initial BAL is 0.0226x 0.1509, but this level is
decreased by 0.015 each hour. Thus, the actual BAL after x hours is
0.0226x 0.1509 0.015x, or 0.0376x 0.1509. (Check with the
calculator at link 4-4-5 on the Bello Website at mhhe.com/bello.)
Answer
7. 0.0407x 0.1663
PROBLEM 7
Find a polynomial that would approximate the BAL for a female x hours after
consuming 3 ounces of alcohol.
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Chapter 4
4-52
Exponents and Polynomials
Exercises 4.5
Boost your GRADE
at mathzone.com!
In Problems 1–30, add as indicated.
1. (5x 2 2x 5) (7x 2 3x 1)
•
•
•
•
•
2. (3x 2 5x 5) (9x 2 2x 1)
Practice Problems
Self-Tests
Videos
NetTutor
e-Professors
3. (3x 5x 2 1) (7 2x 7x 2 )
4. (3 2x 2 7x) (6 2x 2 5x)
5. (2x 5x 2 2) (3 5x 8x 2 )
6. 3x 2 3x 2 and 4 5x 6x 2
7. 2 5x and 3 x 2 5x
8. 4x 2 6x 2 and 2 5x
9. x 3 2x 3 and 2x 2 x 5
10. x 4 3 2x 3x 3 and 3x 4 2x 2 5 x
11. 6x 3 2x 4 x and 2x 2 5 2x 2x 3
1
1
3
1
12. x3 x2 x and x x3 3x2
2
5
5
2
1 2
3
1
1
2
13. x2 x and x x2 3 5
4
4
5
3
14. 0.3x 0.1 0.4x 2 and 0.1x 2 0.1x 0.6
1
1
1
15. 0.2x 0.3 0.5x2 and x x 2
10 10
10
16.
x 2 5x 2
() 3x 2 7x 2
2x 4
18.
()
3x 4
20.
()
22.
3x 3
17.
2x 1
x 3x 5
19.
2x 2 5x 5
2x
5x 7
21.
x1
x 2x 5
x
23.
3
3x 2 2x 4
()
x 2 4x 7
()
3
5x 4
()
5x 2 3
5x 3x 2 5
5x 3 3x 2
2
() 5x 3
3x 4
x 3 2x 5
3x 4
()
3x 2
3
3
5x 9
7
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4-53
24.
4.5
1
x 3
3
1
x 5
2
1 2 1
x x 1
5
2
2
() x 3
3
26.
28.
30.
25.
x2
1
1
1
x 2 x 8
3
5
3
2
x 3 x 2
8
5
2
4
() 3x 3
x 3
5
2x 4 5x 3 2x 2 3x 5
8x 3
2x 5
4
x
3x 2 x 2
()
6x 3
2x 5
27.
Addition and Subtraction of Polynomials
2
1
x 3 x 2
2
7
6
1
x 3
5x 3
7
5
()
x 2
1
6
1
x 3
7
2
1
x 2 x 3
9
2 3 2 2
() x x 2x 5
7
9
29.
6x 3 2x 2
1
x 3x 3 5x 2 3x
x 3
7x 2
4
() 3x
3x 1
4
3x 4
2x 2
5
4
3
x x 2x 7x 2 5x
2x 4
2x 2
7
5
3
() 7x
2x
2x
5
In Problems 31–50, subtract as indicated.
31. (7x 2 2) (3x 2 5)
32. (8x 2 x) (7x 2 3x)
33. (3x 2 2x 1) (4x 2 2x 5)
34. (3x x 2 1) (5x 1 3x 2)
35. (1 7x 2 2x) (5x 3x 2 7)
36. (7x 3 x 2 x 1) (2x 2 3x 6)
37. (5x 2 2x 5) (3x 3 x 2 5)
38. (3x 2 x 7) (5x 3 5 x 2 2x)
39. (6x 3 2x 2 3x 1) (x 3 x 2 5x 7)
40. (x 3x 2 x 3 9) (8 7x x 2 x 3 )
41.
6x 2 3x 5
() 3x 2 4x 2
42.
7x 2 4x 5
() 9x 2 2x 5
343
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43.
Chapter 4
3x 2 2x 1
() 3x 2 2x 1
4x 3
45.
44.
5x 2
1
() 3x 2 2x 1
2x 5
3x 5x 1
46.
3x 2 5x 2
() x 2x 2
5
2
2x x 6
48.
2
()
3x 3
47.
3
2
()
5x 3
49.
4-54
Exponents and Polynomials
x2
5x 3x 7
6x 3
50.
2
()
x 2 2x 1
() 3x x 2 5x 2
3
()
2x 5
3x 2 x
In Problems 51–55, find the sum of the areas of the shaded rectangles.
51.
52.
x
x
A
x
x
B
2
x
C
D
4
3 A
x
53.
2
x B
x
x
x
A
B
x C
x
x
x
x
D
4
A
4
x
B
4
2x
C
2x
D
2x
3x
54.
x
3
C
x
D
4
x
55.
3x
A
2x
3x
B
5
3x
C
3x
3x
D
3x
APPLICATIONS
56. Annual tuna consumption Based on Census Bureau
estimates, the annual consumption of canned tuna (in
pounds) per person is C(t) 3.87 0.13t, while the
consumption of turkey (in pounds) is T(t) 0.15t 2 0.8t 13, where t is the number of years after 1989.
a. How many pounds of canned tuna and how many
pounds of turkey were consumed per person in
1989?
b. Find the difference between the amount of turkey
consumed and the amount of canned tuna
consumed.
c. Use your answer to part b to find the difference in
the amount of turkey and the amount of canned
tuna you would expect to be consumed in the year
2000.
57. Annual poultry consumption Poultry products consist
of turkey and chicken. Annual chicken consumption
per person (in pounds) is C(t) 0.05t 3 0.7t 2 t 36
and annual turkey consumption (in pounds) is T(t) 0.15t 2 1.6t 9, where t is the number of years
after 1985.
a. Find the total annual poultry consumption per
person.
b. How much poultry was consumed per person in
1995? How much poultry do you predict would be
consumed in 2005?
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Addition and Subtraction of Polynomials
345
58. College costs How much are you paying for tuition
and fees? In a four-year public institution, the amount
T(t) you pay for tuition and fees (in dollars) can be approximated by T(t) 45t2 110t 3356, where t is
the number of years after 2000 (2000 0).
a. What would you predict tuition and fees to be in
2005?
b. The cost of books t years after 2000 can be approximated by B(t) 27.5t 680. What would be the
cost of books in 2005?
c. What polynomial would represent the cost
of tuition and fees and books t years after
2000?
d. What would you predict the cost of tuition and fees
and books would be in 2005?
59. College expenses The three major college expenses
are: tuition and fees, books, and room and board. They
can be approximated, respectively, by:
60. Student loans If you are an undergraduate dependent
student you can apply for a Stafford Loan. The amount
of these loans can be approximated by S(t) 562.5t2 312.5t 2625, where t is between 0 and 2
inclusive. If t is between 3 and 5 inclusive, then S(t) $5500.
a. How much money can you get from a Stafford
Loan the first year?
b. What about the second year?
c. What about the fifth year?
61. College financial aid Assume that you have to pay
tuition and fees, books, and room and board in 2003 but
have your Stafford Loan to decrease expenses. (See
Problems 59 and 60.) Write a polynomial that would
approximate how much you would have to pay t years
after 2000 (t between 0 and 2).
T(t) 45t 2 110t 3356
B(t) 27.5t 680
R(t) 32t 2 200t 4730
where t is the number of years after 2000 (2000 0).
a. Write a polynomial representing the total cost of tuition and fees, books, and room and board t years after
2000.
b. What was the cost of tuition and fees, books, and
room and board in 2000?
c. What would you predict the cost of tuition and
fees, books, and room and board would be in 2005?
SKILL CHECKER
Try the Skill Checker Exercises so you’ll be ready for the next section.
Simplify:
62. (5x 3 ) (2x 4 )
63. (2x 4 ) (3x 5 )
65. 6(y 4)
66. 3(2y 3)
64. 5(x 3)
USING YOUR KNOWLEDGE
Business Polynomials
Polynomials are also used in business and economics. For example, the revenue R may be obtained by subtracting the cost C
of the merchandise from its selling price S. In symbols, this is
RSC
Now the cost C of the merchandise is made up of two parts: the variable cost per item and the fixed cost. For example, if you
decide to manufacture Frisbees™, you might spend $2 per Frisbee in materials, labor, and so forth. In addition, you might
have $100 of fixed expenses. Then the cost for manufacturing x Frisbees is
cost per
fixed
Cost C of merchandise is Frisbee and expenses.
14444244443 { 123 123 14243
C
2x
100
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Chapter 4
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Exponents and Polynomials
If x Frisbees are then sold for $3 each, the total selling price S is 3x, and the revenue R would be
RSC
3x (2x 100)
3x 2x 100
x 100
Thus if the selling price S is $3 per Frisbee, the variable costs are $2 per Frisbee, and the fixed expenses are $100, the revenue
after manufacturing x Frisbees is given by
R x 100
In Problems 67–69, find the revenue R for the given cost C and selling price S.
67. C 3x 50; S 4x
68. C 6x 100; S 8x
70. In Problem 68, how many items were manufactured if
the revenue was zero?
69. C 7x; S 9x
71. If the merchant of Problem 68 suffered a $40 loss
($40 revenue), how many items were produced?
WRITE ON
72. Write the procedure you use to add polynomials.
73. Write the procedure you use to subtract polynomials.
74. Explain the difference between “subtract x 2 3x 5
from 7x 2 2x 9” and “subtract 7x 2 2x 9 from
x 2 3x 5.” What is the answer in each case?
75. List the advantages and disadvantages of adding (or
subtracting) polynomials horizontally or in columns.
MASTERY TEST
If you know how to do these problems, you have learned your lesson!
Add:
76. 3x 3x 2 6 and 5x 2 10 x
77. 5x 8x 2 3 and 3x 2 4 8x
Subtract:
78. 3 4x 2 5x from 9x 2 2x
79. 9 x 3 3x 2 from 10 7x 2 5x 3
80. Add 2x 3 3x 5x 2 2, 6 5x 2x 2,
and 6x 3 2x 8.
81. Find the sum of the areas of the shaded rectangles:
3 C
x
A
3
2x
B
2x
x
x
D
2
82. The number of robberies (per 100,000 population) can
be approximated by R(t) 1.85t 2 19.14t 262,
while the number of aggravated assaults is approximated by A(t) 0.2t 3 4.7t 2 15t 300, where t
is the number of years after 1960.
a. Were there more aggravated assaults or more
robberies per 100,000 in 1960?
b. Find the difference between the number of
aggravated assaults and the number of robberies
per 100,000.
c. What would this difference be in the year 2000?
In 2010?
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4.6
4.6
Multiplication of Polynomials
347
MULTIPLICATION OF POLYNOMIALS
To Succeed, Review How To . . .
1. Multiply expressions (pp. 92, 98,
101, 294, 301).
2. Use the distributive property to
remove parentheses in an expression
(p. 101).
Objectives
A Multiply two monomials.
B Multiply a monomial and a binomial.
C Multiply two binomials using the
FOIL method.
D Solve an application.
GETTING STARTED
Deflections on a Bridge
How much does the beam bend
(deflect) when a car or truck goes
over the bridge? There’s a formula
that can tell us. For a certain beam of
length L, the deflection at a distance x
from one end is given by
(x L)(x 2L)
To multiply these two binomials, we
must first learn how to do several
related types of multiplication.
A
Multiplying Two Monomials
We’ve already multiplied two monomials in Section 4.1. The idea is to use the associative
and commutative properties and the rules of exponents, as shown in the next example.
Web It
For practice problems and a tutorial on multiplying monomials, go to link 4-6-1 on the Bello Website at mhhe.com/bello.
EXAMPLE 1
Multiplying two monomials
Multiply: (3x 2) by (2x 3)
Multiply: (4y 3) by (5y 4)
SOLUTION
(3x 2)(2x 3) (3 2)(x 2 x 3)
6x 23
6x5
Answer
1. 20y7
PROBLEM 1
Use the associative and
commutative properties.
Use the rules of exponents.
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Chapter 4
4-58
Exponents and Polynomials
B
Multiplying a Monomial and a Binomial
In Sections 1.6 and 1.7, we also multiplied a monomial and a binomial. The procedure was
based on the distributive property, as shown next.
EXAMPLE 2
PROBLEM 2
Multiplying a monomial by a binomial
Remove parentheses (simplify):
a. 5(x 2y)
Simplify:
b. (x 2 2x)3x 4
a. 4(a 3b)
b. (a2 3a)4a5
SOLUTION
a. 5(x 2y) 5x 5 2y
5x 10y
b. (x 2 2x)3x 4 x 2 3x 4 2x 3x 4
3 x2 x4 2 3 x x4
Since (a b)c ac bc
3x 6 6x 5
NOTE
You can use the commutative property first and write
(x 2 2x)3x 4 3x 4(x 2 2x)
3x 6 6x 5
C
Web It
(Same answer!)
Multiplying Two Binomials Using the FOIL Method
Another way to multiply (x 2)(x 3) is to use the distributive property a(b c) ab ac. Think of x 2 as a, which makes x like b and 3 like c. Here’s how it’s done.
a (b c) a b a c
678 } }
678} 678}
(x 2) (x 3) (x 2)x (x 2)3
To learn about four different
methods of multiplying binomials, go to link 4-6-2 on the
Bello Website at mhhe.com/
bello.
123
x x 123
2 x 123
x 3 123
23
x 2 2x 3x 6
14243
x2 5x
6
Similarly,
(x 3)(x 5) (x 3)x (x 3)5
123
x x (3)
x 123
x 5 (3)
5
14243
14243
x2 x
2
3x 5x 1442443
2x
Can you see a pattern developing? Look at the answers:
(x 2)(x 3) x 2 5x 6
Answers
2. a. 4a 12b
b. 4a7 12a6
(x 3)(x 5) x 2 2x 15
15
15
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4.6
Multiplication of Polynomials
349
It seems that the first term in each answer (x2) is obtained by multiplying the first terms in
the factors (x and x). Similarly, the last terms (6 and 15) are obtained by multiplying the
last terms (2 3 and 3 5). Here’s how it works so far:
Need the
middle term
xx
(x 2)(x 3) x 2 _____ 6
Web It
23
Need the
middle term
To practice FOIL with your
very own polynomials, go to
link 4-6-3 on the Bello Website at mhhe.com/bello.
xx
(x 3)(x 5) x 2 _____ 15
3 5
But what about the middle terms? In (x 2)(x 3), the middle term is obtained by
adding 3x and 2x, which is the same as the result we got when we multiplied the outer
terms (x and 3) and added the product of the inner terms (2 and x). Here’s a diagram that
shows how the middle term is obtained:
Outer terms
x3
(x 2)(x 3) x 2 3x 2x 6
2x
Inner terms
Outer terms
x5
(x 3)(x 5) x 2 5x 3x 15
3 x
Inner terms
Do you see how it works now? Here is a summary of this method.
PROCEDURE
FOIL Method for Multiplying Binomials
First terms are multiplied first.
Outer terms are multiplied second.
Inner terms are multiplied third.
Last terms are multiplied last.
Of course, we call this method the FOIL method. We shall do one more example, step by
step, to give you additional practice.
F
O
I
L
(x 7)(x 4) → x 2
(x 7)(x 4) → x 2 4x
(x 7)(x 4) → x 2 4x 7x
(x 7)(x 4) x 2 4x 7x 28
x 2 3x 28
First: x x
Outer: 4 x
Inner: 7 x
Last: 7 (4)
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Chapter 4
EXAMPLE 3
4-60
Exponents and Polynomials
PROBLEM 3
Using FOIL to multiply two binomials
Find:
Find:
a. (x 5)(x 2)
a. (a 4)(a 3)
b. (x 4)(x 3)
b. (a 5)(a 4)
SOLUTION
(First)
F
a. (x 5)(x 2) x x
123
x2
b. (x 4)(x 3) x x
123
x2
(Outer) (Inner)
O
I
(Last)
L
2x 5x 5 2
1442443
123
3x
10
3x 4x 4 3
1442443
123
x
12
As in the case of arithmetic, we can use the ideas we’ve just discussed to do more
complicated problems. Thus we can use the FOIL method to multiply expressions such as
(2x 5) and (3x 4). We proceed as before; just remember your laws of exponents and
the FOIL sequence.
EXAMPLE 4
PROBLEM 4
Using FOIL to multiply two binomials
Find:
Find:
a. (2x 5)(3x 4)
a. (3a 5)(2a 3)
b. (3x 2)(5x 1)
b. (2a 3)(4a 1)
SOLUTION
(First)
F
(Outer)
O
(Inner)
I
(Last)
L
a. (2x 5)(3x 4) (2x)(3x) (2x)(4) 5(3x) (5)(4)
123
14243
123
123
6x 2
8x
15x 20
144424443
6x 2
7x
20
F
O
b. (3x 2)(5x 1) (3x)(5x) 123
15x 2
15x 2
I
3x(1) 2(5x) 14243
123
3x
10x 1442443
13x
L
2(1)
123
2
2
Does FOIL work when the binomials to be multiplied contain more than one variable?
Fortunately, yes. Again, just remember the sequence and the laws of exponents. For example, to multiply (3x 2y) by (2x 5y), we proceed as follows:
F
Answers
3. a. a2 a 12
b. a2 a 20
4. a. 6a 2 a 15
b. 8a2 14a 3
O
I
L
(2x 5y)(3x 2y) (2x)(3x) (2x)(2y) (5y)(3x) (5y)(2y)
123
123
123
123
6x 2 4xy 15xy 10y 2
1442443
6x 2
19xy
10y 2
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4-61
4.6
EXAMPLE 5
Multiplying binomials involving two variables
Find:
Multiplication of Polynomials
351
PROBLEM 5
Find:
a. (5x 2y)(2x 3y)
a. (4a 3b)(3a 5b)
b. (3x y)(4x 3y)
b. (2a b)(3a 4b)
SOLUTION
F
O
I
L
a. (5x 2y)(2x 3y) (5x)(2x) (5x)(3y) (2y)(2x) (2y)(3y)
123 123 123 123
10x 2 15xy 4xy 6y 2
1442443
10x 2 19xy
6y 2
F
O
I
L
b. (3x y)(4x 3y) (3x)(4x) (3x)(3y) (y)(4x) (y)(3y)
123
14243
14243
14243
12x 2 9xy
4xy 3y2
1442443
12x 2 13xy
3y2
Now one more thing. How do we multiply the expression in the Getting Started?
(x L)(x 2L)
We do it in the next example.
EXAMPLE 6
Multiplying binomials involving two variables
Perform the indicated operation:
PROBLEM 6
Perform the indicated operation:
(x L)(x 2L)
(y 2L)(y 3L)
SOLUTION
F
O
I
L
(x L)(x 2L) x x (x)(2L) (L)(x) (L)(2L)
123 14243 14243 14243
x2 2xL xL 2L2
1442443
x2 3xL
2L2
D
Answers
5. a. 12a2 29ab 15b2
b. 6a2 11ab 4b2
6. y 2 5Ly 6L2
Solving an Application
Suppose we wish to find out how much is spent annually on hospital care. According to
the American Hospital Association, average daily room charges can be approximated by
C(t) 160 14t (dollars), where t is the number of years after 1990. On the other hand,
the U.S. National Health Center for Health Statistics indicated that the average stay (in
days) in the hospital can be approximated by D(t) 7 0.2t, where t is the number of
years after 1990. The amount spent annually on hospital care is given by
Cost per day Number of days C(t) D(t)
We find this product next.
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Chapter 4
EXAMPLE 7
4-62
Exponents and Polynomials
PROBLEM 7
Getting well in U.S. hospitals
Find: C(t) D(t) (160 14t)(7 0.2t)
SOLUTION
Suppose that at a certain hospital, the
cost per day is (200 15t) and the
average stay (in days) is (10 0.1t).
What is the amount spent annually at
this hospital?
We use the FOIL method:
F
O
I
L
(160 14t)(7 0.2t) 160 7 160 (0.2t) 14t 7 14t (0.2t)
123 1442443 123
1442443
1120 32t
98t 2.8t 2
1442443
1120 66t
2.8t 2
Thus the total amount spent annually on hospital care is 1120 66t 2.8t2.
Can you calculate what this amount was for 2000? What will it be for the year
2010?
Some students prefer a grid method to multiply polynomials. Thus, to do Example 4(a), (2x 5) (3x 4), create a grid separated into four compartments. Place the term
(2x 5) at the top and the term (3x 4) on the side of the grid. Multiply the rows and
columns of the grid as shown. (After you get some practice, you can skip the initial step and
write 6x 2, 15x, 8x, and 20 in the grid.)
2x
5
3x
3x 2x
6x 2
3x 5
15x
4
4 2x
8x
4 5
20
Finish by writing the results of each of the grid boxes:
6x 2 15x 8x 20
And combining like terms:
6x 2 7x 20
You can try using this technique in the margin problems or in the exercises!
Calculate It
Checking Equivalency
In the Section 4.1 Calculate It, we agreed that two expressions are equivalent if their graphs
are identical. Thus, to check Example 1 we have to check that (3x 2)(2x 3) 6x 5. Let
Y1 (3x 2)(2x 3) and Y2 6x 5. Press
and the graph shown here will appear. To confirm the
result numerically, press
and you get the result in the table.
X
0
1
2
3
4
5
6
Y1
0
-6
-192
-1458
-6144
-18750
-46656
Y2
0
-6
-192
-1458
-6144
-18750
-46656
X=0
You can check the rest of the examples except Examples 5 and 6. Why?
Answer
7. 1.5t 2 130t 2000
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4-63
4.6
Multiplication of Polynomials
Exercises 4.6
Boost your GRADE
at mathzone.com!
In Problems 1–6, find the product.
1. (5x 3)(9x 2)
2. (8x 4)(9x 3)
3. (2x)(5x 2)
4. (3y 2)(4y 3)
5. (2y 2)(3y)
6. (5z)(3z)
•
•
•
•
•
Practice Problems
Self-Tests
Videos
NetTutor
e-Professors
In Problems 7–20, remove parentheses (simplify).
7. 3(x y)
8. 5(2x y)
9. 5(2x y)
10. 4(3x 4y)
11. 4x(2x 3)
12. 6x(5x 3)
13. (x 2 4x)x 3
14. (x 2 2x)x 2
15. (x x 2)4x
16. (x 3x 2)5x
17. (x y)3x
18. (x 2y)5x 2
19. (2x 3y)(4y 2)
20. (3x 2 4y)(5y3)
In Problems 21–56, use the FOIL method to perform the indicated operation.
21. (x 1)(x 2)
22. (y 3)(y 8)
23. (y 4)(y 9)
24. (y 6)(y 5)
25. (x 7)(x 2)
26. (z 2)(z 9)
27. (x 3)(x 9)
28. (x 2)(x 11)
29. (y 3)(y 3)
30. (y 4)(y 4)
31. (2x 1)(3x 2)
32. (4x 3)(3x 5)
33. (3y 5)(2y 3)
34. (4y 1)(3y 4)
35. (5z 1)(2z 9)
36. (2z 7)(3z 1)
37. (2x 4)(3x 11)
38. (5x 1)(2x 1)
39. (4z 1)(4z 1)
40. (3z 2)(3z 2)
41. (3x y)(2x 3y)
42. (4x z)(3x 2z)
43. (2x 3y)(x y)
44. (3x 2y)(x 5y)
45. (5z y)(2z 3y)
46. (2z 5y)(3z 2y)
47. (3x 2z)(4x z)
48. (2x 3z)(5x z)
49. (2x 3y)(2x 3y)
50. (3x 5y)(3x 5y)
51. (3 4x)(2 3x)
52. (2 3x)(3 2x)
53. (2 3x)(3 x)
54. (3 2x)(2 x)
55. (2 5x)(4 2x)
56. (3 5x)(2 3x)
353
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4-64
Exponents and Polynomials
APPLICATIONS
57. Area of a rectangle The area A of a
rectangle is obtained by multiplying
its length L by its width W; that is,
A LW. Find the area of the rectangle shown in the figure.
x2
x5
58. Area of a rectangle Use the formula in Problem 57 to
find the area of a rectangle of width x 4 and length
x 3.
59. Height of a thrown object The height reached by an 60. Resistance The resistance R of a resistor varies
object t seconds after being thrown upward with a
with the temperature T according to the equation
velocity of 96 feet per second is given by 16t(6 t).
R (T 100)(T 20). Use the distributive property to
Use the distributive property to simplify this expression.
simplify this expression.
61. Gas property expression In chemistry, when V is the
volume and P is the pressure of a certain gas, we find the
expression (V2 V1)(CP PR), where C and R are
constants. Use the distributive property to simplify this
expression.
The garage shown is 40 feet by 20 feet. You want to convert
it to a bigger garage with two storage areas, S1 and S2.
20
62. What is the area of the current garage?
5
Garage
extension
64. Calculate the areas of S1, S2, and S3, and write your
answers in the appropriate places in the diagram.
S3
40
65. Determine the total area of the new garage by adding
the area of the original garage to the areas of S1, S2, and
S3; that is, add the answers you obtained in Problems 62
and 64.
Garage
40 20
8
66. Is the area of the new garage (Problem 63) the same as
the answer in Problem 65?
S2
S1
63. If you extend the long side by 8 feet and the short side
by 5 feet, what is the area of the new garage?
Storage areas
y
20
Garage
extension
40
S3
S1
b. Find the area of S2.
c. Find the area of S3.
68. The area of the new garage is (40 x)(20 y). Simplify this expression.
Garage
40 20
x
67. If you are not sure how big you want the storage rooms,
extend the long side of the garage by x feet and the short
side by y feet.
a. Find the area of S1.
69. Add the areas of S1, S2, S3, and the area of the original
garage. Is the answer the same as the one you obtained
in Problem 68?
S2
Storage areas
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4.6
Multiplication of Polynomials
355
SKILL CHECKER
Try the Skill Checker Exercises so you’ll be ready for the next section.
Find:
70. (4y)2
71. (3x)2
73. (3A)2
74. A(A)
72. (A)2
USING YOUR KNOWLEDGE
Profitable Polynomials
Do you know how to find the profit made in a certain business? The profit P is the difference between the revenue R and the
expense E of doing business. In symbols,
PRE
Of course, R depends on the number n of items sold and their price p (in dollars); that is,
R np
Thus
P np E
75. If n 3p 60 and E 5p 100, find P.
76. If n 2p 50 and E 3p 300, find P.
77. In Problem 75, if the price was $2, what was the profit
P?
78. In Problem 76, if the price was $10, what was the profit
P?
WRITE ON
79. Will the product of two monomials always be a monomial? Explain.
80. If you multiply a monomial and a binomial, will you
ever get a trinomial? Explain.
81. Will the product of two binomials (after combining like
terms) always be a trinomial? Explain.
82. Multiply:
(x 1)(x 1) (y 2)(y 2) (z 3)(z 3) What is the pattern?
MASTERY TEST
If you know how to do these problems, you have learned your lesson!
Multiply:
83. (7x 4)(5x 2)
84. (8a3)(5a5)
85. (x 7)(x 3)
86. (x 2)(x 8)
87. (3x 4)(3x 1)
88. (4x 3y)(3x 2y)
89. (5x 2y)(2x 3y)
90. (x L)(x 3L)
91. Simplify 6(x 3y).
92. Simplify (x 3 5x)(4x 5).