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348 CHAPTER 6 Rational Expressions
105. y^2 n^ + 9 10 y
# y^
n (^) - 3 y^4 n^ - 81
106.
y^4 n^ - 16 y^2 n^ + 4
# 6 y
y n^ + 2
107. y^2 n^ - y n^ - 2 2 y n^ - 4 , y^2 n^ - 1 1 + y n
108. y^2 n^ + 7 y n^ + 10 10 ,^
y^2 n^ + 4 y n^ + 4 5 y n^ + 25
Simplify. Assume that no denominator is 0.
99. p x^ - 4 4 - p x^ 100.^
3 + q n q n^ + 3
101. x^
n (^) + 4 x^2 n^ - 16
102. x
2 k (^) - 9 3 + x k Perform the indicated operation. Write all answers in lowest terms.
103. x^2 n^ - 4 7 x
# 14 x
3 x n^ - 2
104. x
2 n (^) + 4 x n (^) + 4 4 x - 3
# 8 x
(^2) - 6 x x n^ + 2
OBJECTIVE
1 Adding or Subtracting Rational Expressions with a
Common Denominator
Rational expressions, like rational numbers, can be added or subtracted. We add or subtract rational expressions in the same way that we add or subtract rational numbers (fractions).
OBJECTIVES 1 Add or Subtract Rational Expressions with a Common Denominator. 2 Identify the Least Common Denominator (LCD) of Two or More Rational Expressions. 3 Add or Subtract Rational Expressions with Unlike Denominators.
6.2 Adding and Subtracting Rational Expressions
Adding or Subtracting Rational Expressions with a Common Denominator
If
P
Q
and
R
Q
are rational expressions, then
P Q
R
Q
P + R
Q
and
P
Q
R
Q
P - R
Q
To add or subtract rational expressions with a common denominator, add or subtract the numerators and write the sum or difference over the common denominator.
E X A M P L E 1 Add or subtract.
a. x 4
5 x 4 b.
7 z^2
x 7 z^2 c. x^2 x + 7
x + 7 d. x 3 y^2
x + 1 3 y^2
Solution The rational expressions have common denominators, so add or subtract
their numerators and place the sum or difference over their common denominator.
a. x 4
5 x 4
x + 5 x 4
6 x 4
3 x 2
Add the numerators and write the result over the common denominator.
b.
7 z^2
x 7 z^2
5 + x 7 z^2
c. x^2 x + 7
x + 7
x^2 - 49 x + 7
Subtract the numerators and write the result over the common denominator.
=
1 x + 72 1 x - 72 x + 7 Factor the numerator.
= x - 7 Simplify.
d. x 3 y^2
x + 1 3 y^2
x - 1 x + 12 3 y^2 Subtract the numerators.
x - x - 1 3 y^2 Use the distributive property.
3 y^2 Simplify.
Helpful Hint Very Important: Be sure to insert parentheses here so that the entire numerator is subtracted.
Section 6.2 Adding and Subtracting Rational Expressions 349
PRACTICE 1 Add or subtract.
a.
11 z^2
x 11 z^2 b. x 8
5 x 8 c. x^2 x + 4
x + 4 d. z 2 a^2
z + 3 2 a^2
OBJECTIVE
CONCEPT CHECK Find and correct the error. 3 + 2 y y^2 - 1
y + 3 y^2 - 1
3 + 2 y - y + 3 y^2 - 1
=
y + 6 y^2 - 1
2 Identifying the Least Common Denominator (LCD)
of Rational Expressions
To add or subtract rational expressions with unlike denominators, first write the ratio- nal expressions as equivalent rational expressions with a common denominator. The least common denominator (LCD) is usually the easiest common denomina- tor to work with. The LCD of a list of rational expressions is a polynomial of least degree whose factors include the denominator factors in the list. Use the following steps to find the LCD.
Finding the Least Common Denominator (LCD) Step 1. Factor each denominator completely. Step 2. The LCD is the product of all unique factors, each raised to a power equal to the greatest number of times that the factor appears in any factored denominator.
E X A M P L E 2 Find the LCD of the rational expressions in each list.
a.
3 x^5 y^2
3 z 5 xy^3
b.
z + 1
z z - 1
c. m - 1 m^2 - 25
2 m 2 m^2 - 9 m - 5
m^2 - 10 m + 25
d. x x^2 - 4
6 - 3 x
Solution
a. First we factor each denominator. 3 x^5 y^2 = 3 #^ x^5 #^ y^2 5 xy^3 = 5 #^ x #^ y^3 LCD = 3 #^5 #^ x^5 #^ y^3 = 15 x^5 y^3 b. The denominators z + 1 and z - 1 do not factor further. Thus, LCD = 1 z + 121 z - 12 c. We first factor each denominator. m^2 - 25 = 1 m + 521 m - 52 2 m^2 - 9 m - 5 = 12 m + 121 m - 52 m^2 - 10 m + 25 = 1 m - 521 m - 52 LCD = 1 m + 5212 m + 121 m - 522
Answer to Concept Check: 3 + 2 y y^2 - 1
- y + 3 y^2 - 1 = 3 + 2 y - y - 3 y^2 - 1 = y y^2 - 1
Helpful Hint The greatest power of x is 5, so we have a factor of x^5. The great- est power of y is 3, so we have a factor of y^3.
Section 6.2 Adding and Subtracting Rational Expressions 351
PRACTICE
PRACTICE
b. The LCD is the product of the two denominators: 1 x + 221 x - 22. 3 x + 2
2 x x - 2
3 #^1 x - 22 1 x + 22 #^1 x - 22
2 x #^1 x + 22 1 x - 22 #^1 x + 22
Write equivalent rational expressions.
= 3 x - 6 1 x + 221 x - 22
2 x^2 + 4 x 1 x + 221 x - 22 Multiply in the numerators.
3 x - 6 + 2 x^2 + 4 x 1 x + 221 x - 22 Add the numerators.
2 x^2 + 7 x - 6 1 x + 221 x - 22 Simplify the numerator.
c. The LCD is either x - 1 or 1 - x. To get a common denominator of x - 1, we factor - 1 from the denominator of the second rational expression. 2 x - 6 x - 1
1 - x
2 x - 6 x - 1
- 11 x - 12 Write 1 - x as - 11 x - 12.
2 x - 6 x - 1
- 1 #^4 x - 1 Write 4
- 11 x - 12 as -^1
(^4)
x - 1 .
2 x - 6 - 1 - 42 x - 1 Combine the numerators.
2 x - 6 + 4 x - 1 Simplify.
2 x - 2 x - 1
=
21 x - 12 x - 1 Factor.
= 2 Simplest form
3 Perform the indicated operation.
a.
p^3 q
5 p^4 q
b.
y + 3
5 y y - 3 c. 3 z - 18 z - 5
5 - z
E X A M P L E 4 Subtract
5 k k^2 - 4
k^2 + k - 2
Solution
5 k k^2 - 4
k^2 + k - 2
5 k 1 k + 221 k - 22
1 k + 221 k - 12 The LCD is 1 k + 221 k - 221 k - 12. We write equivalent rational expressions with the LCD as denominators. 5 k 1 k + 221 k - 22
1 k + 221 k - 12
=
5 k #^1 k - 12 1 k + 221 k - 22 #^1 k - 12
2 #^1 k - 22 1 k + 221 k - 12 #^1 k - 22
Write equivalent rational expressions.
= 5 k^2 - 5 k 1 k + 221 k - 221 k - 12
2 k - 4 1 k + 221 k - 221 k - 12
Multiply in the numerators.
= 5 k^2 - 5 k - 2 k + 4 1 k + 221 k - 221 k - 12
Subtract the numerators.
= 5 k^2 - 7 k + 4 1 k + 221 k - 221 k - 12 Simplify.
4 Subtract t t^2 - 25
t^2 - 3 t - 10
Helpful Hint Very Important: Because we are subtracting, notice the sign change on 4.
Factor each denominator to find the LCD.
352 CHAPTER 6 Rational Expressions
E X A M P L E 5 Add
2 x - 1 2 x^2 - 9 x - 5
x + 3 6 x^2 - x - 2
Solution
2 x - 1 2 x^2 - 9 x - 5
x + 3 6 x^2 - x - 2
2 x - 1 12 x + 121 x - 52
x + 3 12 x + 1213 x - 22
Factor the denominators. The LCD is 12 x + 121 x - 5213 x - 22.
=
12 x - 12 #^13 x - 22 12 x + 121 x - 52 #^13 x - 22
1 x + 32 #^1 x - 52 12 x + 1213 x - 22 #^1 x - 52
= 6 x^2 - 7 x + 2 12 x + 121 x - 5213 x - 22
x^2 - 2 x - 15 12 x + 121 x - 5213 x - 22
= 6 x^2 - 7 x + 2 + x^2 - 2 x - 15 12 x + 121 x - 5213 x - 22
= 7 x^2 - 9 x - 13 12 x + 121 x - 5213 x - 22
Multiply in the numerators.
Add the numerators.
Simplify.
PRACTICE 5 Add 2 x + 3 3 x^2 - 5 x - 2
x - 6 6 x^2 - 13 x - 5
E X A M P L E 6 Perform each indicated operation.
x - 1
10 x x^2 - 1
x + 1
Solution
x - 1
10 x x^2 - 1
x + 1
x - 1
10 x 1 x - 121 x + 12
x + 1
Factor the denominators. The LCD is 1 x - 121 x + 12.
=
7 #^1 x + 12 1 x - 12 #^1 x + 12
10 x 1 x - 121 x + 12
5 #^1 x - 12 1 x + 12 #^1 x - 12
= 7 x + 7 1 x - 121 x + 12
10 x 1 x - 121 x + 12
5 x - 5 1 x + 121 x - 12
= 7 x + 7 + 10 x - 5 x + 5 1 x - 121 x + 12
= 12 x + 12 1 x - 121 x + 12
=
12 1 x + 12 1 x - 12 1 x + 12
=
x - 1
Multiply in the numerators.
Add and subtract the numerators.
Simplify.
Factor the numerator.
Divide out common factors.
PRACTICE 6 Perform each indicated operation. 2 x - 2
3 x x^2 - x - 2
x + 1
354 CHAPTER 6 Rational Expressions
48. x x^2 - 7 x + 6
- x + 4 3 x^2 - 2 x - 1 49.^2 a^2 + 2 a + 1
3 a^2 - 1 50.^9 x^ +^2 3 x^2 - 2 x - 8
7 3 x^2 + x - 4 MIXED PRACTICE Add or subtract as indicated. If possible, simplify your answer. See Examples 1 through 6. 51.^4 3 x^2 y^3
5 3 x^2 y^3 52.^7 2 xy^4
1 2 xy^4
53.^13 x 2^ x -^5 - 13 x 2^ x +^5
54. 17 x + 4 4 x -^
17 x - 4 4 x
55. (^) 2 x +^3 10 + (^3) x +^8 56.^10 3 x - 3 + 1 7 x - 7 57. -^2 x^2 - 3 x - 1 x^3 - 3 x^2 58. -^3 2 a + 8 - 8 a^2 + 4 a 59. ab a^2 - b^2
60. x 25 - x^2
2 3 x - 15
61.^5 x^2 - 4 - 3 x^2 + 4 x + 4 62.^3 z z^2 - 9 - 2 3 - z 63.^3 x 2 x^2 - 11 x + 5
64.^2 x 3 x^2 - 13 x + 4 + 5 x^2 - 2 x - 8 65. (^) x^2 + 1 - (^3) x^3 x + 3 + (^2) x^1 + 2
66. 5 3 x - 6 -^
x x - 2 +^
3 + 2 x 5 x - 10
67. (^) x^3 + 3 + 5 x^2 + 6 x + 9 - x x^2 - 9 68. x^ +^2 x^2 - 2 x - 3 + x x - 3 - x x + 1
69. x x^2 - 9
3 x^2 - 6 x + 9
Find the LCD of the rational expressions in each list. See Example 2.
15.^27 , 53 x 16. (^) 54 y , 3 4 y^2 17.^3 x , (^) x +^2 1 18. (^) 25 x , (^2 7) + x
19. 12 x + 7 ,^
8 x - 7 20.^
1 2 x - 1 ,^
8 2 x + 1
21. (^) 3 x^5 + 6 , (^2) x^2 - x 4 22. (^) 3 a^2 + 9 , (^5) a -^5 23.^2 a a^2 - b^2 , 1 a^2 - 2 ab + b^2 24.^2 a a^2 + 8 a + 16 , 7 a a^2 + a - 12 25. x x^2 - 9
,^5 x , 7 12 - 4 x
26.^9 x^2 - 25
, 1 50 - 10 x ,^6 x Add or subtract as indicated. Simplify each answer. See Examples 3a and 3b.
27. (^) 34 x + (^23) x 28.^107 x + (^25) x
29. 5 2 y^2
4 11 x^4
- 1 4 x^2 31. x x^ - +^34 - x x^ +-^24 32. x x^ - -^15 - x x^ ++^25
33. 1 x - 5 -^
19 - 2 x 1 x - 521 x + 42 34.^
4 x - 2 1 x - 521 x + 42 -^
2 x + 4
Perform the indicated operation. If possible, simplify your answer. See Example 3c.
35. 1 a - b +^
1 b - a 36.^
1 a - 3 -^
1 3 - a
37. x^ +^1 1 - x + 1 x - 1 38.^5 1 - x - 1 x - 1 39.^5 x - 2 + x^ +^4 2 - x 40.^3 5 - x + x^ +^2 x - 5 Perform each indicated operation. If possible, simplify your answer. See Examples 4 through 6.
41. y + 1 y^2 - 6 y + 8
- 3 y^2 - 16 42. x^ +^2 x^2 - 36
- x x^2 + 9 x + 18 43. x^ +^4 3 x^2 + 11 x + 6
- x - 1 x^2 + 4 x + 3 46. a a^2 + 10 a + 25
- 4 -^ a a^2 + 6 a + 5 47. x x^2 - 8 x + 7
- x^ +^2 2 x^2 - 9 x - 35
Section 6.2 Adding and Subtracting Rational Expressions 355
70. 3 x^2 - 9
1 x + 3 71. a (^1) x + 23 b - a (^1) x - 23 b 72. a 1 2
MIXED PRACTICE (SECTIONS 6.1, 6.2)
Perform the indicated operation. If possible, simplify your answer.
73. a 23 - (^1) x b #^ a 3 x +^
1 2 b
74. a 2 3 - 1 x b , a 3 x + 1 2 b 75. a 2 a 3 b
2 , a a
2 a + 1
- 1 a + 1 b 76. a x^ 2 + x^^2 - x^ 2 - x^^2 b #^ a 54 x b
2
77. a 2 x 3 b
2 , a x 3 b
2
78. a 2 x 3 b
2
(^) a 3
x b
2
79. a (^) x + x 1 - (^) x - x 1 b , (^2) x x + 2
80. x 2 x + 2 ,^ a^
x x + 1 +^
x x - 1 b
81.^4 x
(^) a 2
x + 2
- 2 x - 2 b 82. (^) x +^1 1 #^ a (^5) x + (^) x^2 - 3 b
REVIEW AND PREVIEW
Use the distributive property to multiply the following. See Section 1.4.
83. 12 a 2 3 + 1 6 b 84. 14 a 1 7 + 3 14 b 85. x^2 a 4 x^2 + 1 b 86. 5 y^2 a 1 y^2 - 15 b
Find each root. See Section 1.3.
87. 2100 88. 225 89. 23 8 90. 23 27 91. 24 81 92. 24 16
Use the Pythagorean theorem to find the unknown length in each right triangle. See Section 5.8.
93.
4 meters
3 meters
94.
24 feet
7 feet
CONCEPT EXTENSIONS
Find and correct each error. See the Concept Check in this section.
95.^2 x^ -^3 x^2 + 1 - x^ -^6 x^2 + 1
= 2 x^ -^3 -^ x^ -^6 x^2 + 1 = x^ -^9 x^2 + 1
96.^7 x + 7 - x^ +^3 x + 7 = 7 -^ x^ -^3 1 x + 722 = - x + 4 1 x + 722 97. Find the perimeter and the area of the square.
x x 5 ft
98. Find the perimeter of the quadrilateral.
x^2 2 x x 1 cm
x^2 8 x 1 cm
5 x 1 cm
3 x x 1 cm
99. When is the LCD of two rational expressions equal to the product of their denominators? a Hint: What is the LCD of 1 x and 7 x + 5 ? b 100. When is the LCD of two rational expressions with different denominators equal to one of the denominators? a Hint: What is the LCD of (^) x^3 + x 2 and 7 x^ +^1 1 x + 223 ? b 101. In your own words, explain how to add rational expressions with different denominators. 102. In your own words, explain how to multiply rational expressions. 103. In your own words, explain how to divide rational expressions. 104. In your own words, explain how to subtract rational expres- sions with different denominators. Perform each indicated operation. (Hint: First write each expres- sion with positive exponents.) 105. x -^1 + 12 x 2 -^1 106. y -^1 + 14 y 2 -^1 107. 4 x -^2 - 3 x -^1 108. 14 x 2 -^2 - 13 x 2 -^1 Use a graphing calculator to support the results of each exercise. 109. Exercise 7 110. Exercise 8
