Download Multiplication and Division

Annual cost
(hundred thousand dollars)
6.2
a) Use the accompanying graph to estimate the cost for
removing 90% and 95% of the toxic chemicals.
b) Use the formula to determine the cost for removing
99.5% of the toxic chemicals.
c) What happens to the cost as the percentage of pollutants removed approaches 100%?
a) $50,000, $100,000
b) $1,000,000
c) Gets larger and larger without bound
6.2
In this
section
Multiplication and Division
(6-11)
313
5
4
3
2
1
0
90 91 92 93 94 95 96 97 98 99
Percentage of chemicals removed
FIGURE FOR EXERCISE 104
MULTIPLICATION AND DIVISION
In Section 6.1 you learned to reduce rational expressions in the same way that we
reduce rational numbers. In this section we will multiply and divide rational
expressions using the same procedures that we use for rational numbers.
●
Multiplication of Rational
Numbers
Multiplication of Rational Numbers
●
Multiplication of Rational
Expressions
Two rational numbers are multiplied by multiplying their numerators and multiplying their denominators.
●
Division of Rational
Numbers
●
Division of Rational
Expressions
●
Applications
Multiplication of Rational Numbers
If b 0 and d 0, then
a c ac
.
b d bd
E X A M P L E
1
Multiplying rational numbers
Find the product
helpful
hint
Did you know that the line
separating the numerator and
denominator in a fraction is
called the vinculum?
6 14
.
7 15
Solution
The product is found by multiplying the numerators and multiplying the denominators:
6 14
84
7 15 105
21 4
Factor the numerator and denominator.
21 5
4
Divide out the GCF 21.
5
The reducing that we did after multiplying is easier to do before multiplying. First
factor all terms, reduce, and then multiply:
6 14 2 3 2 7
7
3 5
7 15
4
5
■
314
(6-12)
Chapter 6
Rational Expressions
Multiplication of Rational Expressions
We multiply rational expressions in the same way we multiply rational numbers. As
with rational numbers, we can factor, reduce, and then multiply.
E X A M P L E
2
Multiplying rational expressions
Find the indicated products.
9x 10y
a) 5y 3xy
8xy4 15z
b) 3z3
2x5y3
Solution
3 3x 2 5y
9x 10y
a) Factor.
5y
3xy
5y 3xy
6
y
8xy4 15z
2 2 2xy4 3 5z
b) 3z3
3z3
2x5y3
2x5y3
20xy4z
z3x5y3
20y
z2x4
E X A M P L E
3
Factor.
Reduce.
Quotient rule
■
Multiplying rational expressions
Find the indicated products.
2x 2y
2x
a) 4
x2 y2
x2 7x 12
x
b) 2x 6
x2 16
ab
8a2
c) 2
6a
a 2ab b2
study
tip
We are all creatures of habit.
When you find a place in
which you study successfully,
stick with it. Using the same
place for studying will help
you to concentrate and associate the place with good
studying.
Solution
2x 2y
2(x y)
2 x
2x
a) 2
2 4
2
2
(x
y)
(x y)
x y
x
xy
Factor.
Reduce.
x2 7x 12
(x 3) (x 4)
x
x
b) 2
2x 6
2(x 3)
x 16
(x 4) (x 4)
x
2(x 4)
x
2x 8
Factor.
Reduce.
6.2
Multiplication and Division
ab
a b 2 4a2
8a2
c) 6a
a2 2ab b2 2 3a (a b)2
4a
3(a b)
4a
3a 3b
(6-13)
315
Factor.
Reduce.
■
Division of Rational Numbers
Division of rational numbers can be accomplished by multiplying by the reciprocal
of the divisor.
Division of Rational Numbers
If b 0, c 0, and d 0, then
a
c
a d
.
b d b c
E X A M P L E
4
Dividing rational numbers
Find each quotient.
1
a) 5 2
6
3
b) 7 14
Solution
1
a) 5 5 2 10
2
6
6 14 2 3 2 7
3
b) 4
7 14 7 3
7
3
■
Division of Rational Expressions
We divide rational expressions in the same way we divide rational numbers: Invert
the divisor and multiply.
E X A M P L E
helpful
5
hint
A doctor told a nurse to give a
patient half of the usual dose
of certain medicine. The nurse
figured,“dividing in half means
dividing by 1/2 which means
multiply by 2.” So the patient
got four times the prescribed
amount and died (true story).
There is a big difference between dividing a quantity in
half and dividing by one-half.
Dividing rational expressions
Find each quotient.
5
5
a) 3x 6x
x2
4 x2
c) 2 x x x2 1
Solution
5
5
5 6x
a) 3x 6x 3x 5
5 2 3x
3x
5
2
x7
b) (2x 2)
2
Invert the divisor and multiply.
Factor.
Divide out the common factors.
316
(6-14)
Chapter 6
Rational Expressions
x7 1
x7
b) (2x 2) 2
2
2 2x
x5
4
Invert and multiply.
Quotient rule
4 x2
x2
4 x2 x2 1
c) x2 x x2 1 x2 x x 2
Invert and multiply.
1
(2 x) (2 x) (x 1)(x 1)
x(x 1)
x2
1(2 x)(x 1)
x
1(x2 x 2)
x
2
x x 2
x
Factor.
2x
1
x2
Simplify.
■
We sometimes write division of rational expressions using the fraction bar. For
example, we can write
ab
3
ab 1
as .
1
3
6
6
No matter how division is expressed, we invert the divisor and multiply.
E X A M P L E
6
Division expressed with a fraction bar
Find each quotient.
x2 1
ab
2
3
a) b) x1
1
3
6
2
a 5
3
c) 2
Solution
ab
ab 1
3
a) 1
3
6
6
ab 6
3
1
a b 2 3
3
1
(a b)2
2a 2b
Rewrite as division.
Invert and multiply.
Factor.
Reduce.
6.2
x2 1
x2 1 x 1
2
b) x 1
2
3
3
x2 1
3
2
x1
(x 1)(x 1)
3
2
x1
3x 3
2
Multiplication and Division
(6-15)
317
Rewrite as division.
Invert and multiply.
Factor.
Reduce.
a2 5
a2 5
3
c) 2 Rewrite as division.
3
2
2
a 5 1 a2 5
4
6
2
■
Applications
We saw in the last section that rational expressions can be used to represent rates.
Note that there are several ways to write rates. For example, miles per hour is
i
written mph, mi/hr, or m
. The last way is best when doing operations with rates
hr
because it helps us reconcile our answers. Notice how hours “cancels” when we
multiply miles per hour and hours in the next example, giving an answer in miles,
as it should be.
E X A M P L E
7
Writing rational expressions
Answer each question with a rational expression.
0
a) Shasta averaged 20
mph for x hours before she had lunch. How many miles
x
did she drive in the first 3 hours after lunch assuming that she continued to
0
average 20
mph?
x
b) If a bathtub can be filled in x minutes, then the rate at which it is filling is
1
tub/min. How much of the tub is filled in 10 minutes?
x
Solution
a) Because R T D, the distance she traveled after lunch is the product of the
rate and time:
600
200 mi
3 hr mi
x hr
x
b) Because R T W, the work completed is the product of the rate and time:
1 tub
10
10 min tub
x min
x
■
318
(6-16)
Chapter 6
Rational Expressions
WARM-UPS
True or false? Explain your answer.
1.
2
3
5
10
3 9. True
2. The product of
x7
3
and
6
7x
is 2.
True
3. Dividing by 2 is equivalent to multiplying by 1.
2
1
3
4. 3 x x for any nonzero number x.
True
False
5. Factoring polynomials is essential in multiplying rational expressions.
True
6. One-half of one-fourth is one-sixth. False
7. One-half divided by three is three-halves. False
8. The quotient of (839 487) and (487 839) is 1. True
9.
10.
6. 2
a
3
a
b
a
3 9 for any value of a.
b
a
1 for any nonzero values of a and b.
True
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. How do you multiply rational numbers?
Rational numbers are multiplied by multiplying their
numerators and their denominators.
2. How do you multiply rational expressions?
Rational expressions are multiplied by multiplying their
numerators and their denominators.
3. What can be done to simplify the process of multiplying
rational numbers or rational expressions?
Reducing can be done before multiplying rational numbers
or expressions.
4. How do you divide rational numbers or rational expressions?
To divide rational expressions, invert the divisor and
multiply.
Perform the indicated operation. See Example 1.
8 35 7
3 8 2
12 51
5. 6. 7. 15 24 9
4 21 7
17 10
25 56
8. 48 35
True
5
6
7
9. 24 20
42
5
18
5
3
10. 35
10
21
2
Perform the indicated operation. See Example 2.
5a 3ab a
3m 35p 5
11. 12. 7p 6mp 2p
12b 55a 44
2x6 21a2 x5
13. 7a5
6x
a3
5z3w 6y5 wy2
14. 3 9y 20z9 6z6
15t3y5
18t8y7
15. 24t5w3y2 7
20w
w4
6x5
16. 22x2y3z 33y3z4
4x7
z3
Perform the indicated operation. See Example 3.
3a 3b
10a
2a
17. 15
a2 b2 a b
b3 b
10
2b2 2
18. 2
5
b b b1
3
19. (x2 6x 9) 3x 9
x3
12
20. (4x2 20x 25) 12x 30
4x 10
16a 8 2a2 a 1 8a 8
21. 5a2 5
4a2 1
5a2 5
6x 18
4x2 4x 1
22. 2
2
2x 5x 3
6x 3
Perform the indicated operation. See Example 4.
2
1
5 15 2
23. 12 30
24. 32 128 25. 5
4
7 14 3
22
22
3 15 1
40
10
26. 27. 12 28. 9 9
81
4
2 10
3
9
Perform the indicated operation. See Example 5.
5x 2 10x 7x
14u 10uv5
4u2
29. 30. 6 3
15v
21 2
3v
7
2
4
8m3
2
m
2
p
p3
31. (12mn2) 6
32. 3 (4pq5) 8
4
n
3n
3q
6q
y6 6y
33. 3
2
6
4 a a2 16
3
34. 5
3
5a 20
6.2
2
x 2 4x 4 (x 2)3
35. x2
8
16
a2 2a 1 a2 1 a2 a
36. 3
a
3a 3
t 2 3t 10
1
37. (4t 8) t 2 25
4t 20
1
w2 7w 12
38. (w2 9) w2 4w
w2 3w
2x 5
39. (2x2 3x 5) x2 1
x1
2
y
1
40. (6y2 y 2) 9y2 12y 4
3y 2
w2 1
w1
1
64. 2 (w 1) w2 2w 1 w 1
2x2 19x 10
4x2 1
65. 1
2
2
x 100
2x 19x 10
2
x3 1 9x 9x 9 x3 2x2 x
66. 2
x2 x
x 1
9x2 9
9 6m m2
m2 9
67. 2 9 6m m m2 mk 3m 3k
Perform the indicated operation.
2x 2y
1
9
x1
50. 49. 3
3
yx
1x
3
ab
a
ab 2
52. 2b 2a 5
2g
3h
54. 1
h
a6b8
2
2mn4 3m5n7
1
59. 2 2
6mn
m n4 9m3n
rt 2
rt 2
60. 2 32 r 2
rt
rt
3x2 16x 5
x2
61. 2
x
9x 1
3x 3w bx bw 6 2b
68. 2
x2 w2
9b
2g
3
(m 3)2
(m 3)(m k)
2
xw
Solve each problem. Answers could be rational expressions. Be
sure to give your answer with appropriate units. See Example 7.
69. Distance. Florence averaged 26.2
mph for the x hours in
x
which she ran the Boston Marathon. If she ran at that same
rate for 1 hour in the Manchac Fun Run, then how many
2
miles did she run at Manchac?
13.1
mi
x
70. Work. Henry sold 120 magazine subscriptions in x 2
days. If he sold at the same rate for another week, then how
many magazines did he sell in the extra week?
840
magazines
x2
71. Area of a rectangle. If the length of a rectangular flag is
x meters and its width is
2
3
5
x
meters, then what is the area of
the rectangle?
5 square meters
1
5
8x
4
56. (18x) 81
9
2a2 20a 8
58. 3a2 15a3 9a2
5
—m
x
xm
FIGURE FOR EXERCISE 71
72. Area of a triangle. If the base of a triangle is 8x 16
1
yards and its height is yards, then what is the area of
x2
the triangle?
4 yd2
1
—
x+2
x2 5x
3x 1
319
x 2 6x 5
x4 5x3
x4
62. x
3
3x 3
a2 2a 4 (a 2)3 a3 8
63. a2 4
2a 4 2a 4
Perform the indicated operation. See Example 6.
x2 4
3m 6n
x 2y
12
8
5
41. 42. 43. 1
x2
3
10
6
4
m 2n
x2
2x 4y
2
2
6a2 6
x2 9
1
5
3
a3
44. 45. 46. 4
5
6a 6
5
a2 1
x2 9
1
a1
15
4a 12
x2 y2
x2 6x 8
47. 48. xy
x2
9
x1
2
9x 9y
x 5x 4
3a 3b 1
51. a
3
b
a 2b
53. 1 a
2
6y
y
55. (2x) 3
x
a3b4 a5b7
57. 2 2ab
ab
(6-17)
Multiplication and Division
yd
8x + 16 yd
FIGURE FOR EXERCISE 72
320
(6-18)
Chapter 6
Rational Expressions
GET TING MORE INVOLVED
6x2 23x 20
24x 29x 4
74. Exploration. Let R and H 2
73. Discussion. Evaluate each expression.
1
a) One-half of b) One-third of 4
c) One-half of
d) One-half of 1
a) 8
4
b) 3
4
4x
3
a) Find R when x 2 and x 3. Find H when x 2 and
x 3.
b) How are these values of R and H related and why?
3
3
11
11
a) R , R , H , H 5
5
23
23
3x
2
3x
d) 4
2x
c) 3
6.3
In this
section
2x 5
.
8x 1
FINDING THE LEAST COMMON
DENOMINATOR
Every rational expression can be written in infinitely many equivalent forms.
Because we can add or subtract only fractions with identical denominators, we must
be able to change the denominator of a fraction. You have already learned how to
change the denominator of a fraction by reducing. In this section you will learn the
opposite of reducing, which is called building up the denominator.
●
Building Up the
Denominator
●
Finding the Least Common
Denominator
Building Up the Denominator
●
Converting to the LCD
To convert the fraction
2
3
into an equivalent fraction with a denominator of 21, we
factor 21 as 21 3 7. Because
numerator and denominator of
2
3
2
3
already has a 3 in the denominator, multiply the
by the missing factor 7 to get a denominator of 21:
2
2 7
14
3
3 7
21
For rational expressions the process is the same. To convert the rational
expression
5
x3
into an equivalent rational expression with a denominator of x2 x 12, first
factor x2 x 12:
x2 x 12 (x 3)(x 4)
From the factorization we can see that the denominator x 3 needs only a factor of
x 4 to have the required denominator. So multiply the numerator and denominator by the missing factor x 4:
5(x 4)
5x 20
5
2
x 3 (x 3)(x 4) x x 12
E X A M P L E
1
Building up the denominator
Build each rational expression into an equivalent rational expression with the
indicated denominator.
3
?
2
?
?
b) c) 3 8
a) 3 12
w wx
3y
12y