Section 7.5
Sums and Differences of Rational Functions 691
7.5 Exercises
In Exercises 1-16, add or subtract the
rational expressions, as indicated, and
simplify your answer. State all restrictions.
1.
2.
1
7x2 − 49x
42
+
x−6
x−6
2x2 − 110
12
−
x−7
7−x
14.
9x2
144x − 576
+
x−8
8−x
15.
3x2 − 12
15
+
x−3
3−x
16.
112x − 441
7x2
−
x−9
x−9
3.
27x − 9x2
162
+
x+3
x+3
In Exercises 17-34, add or subtract the
rational expressions, as indicated, and
simplify your answer. State all restrictions.
4.
2x2 − 28
10x
−
x+2
x+2
17.
5.
4x2 − 8
56
+
x−4
4−x
18.
6.
4x2
36x − 56
−
x−2
x−2
19.
7.
9x2
72x − 63
+
x−1
1−x
20.
9x
45
−
x2 − 25 x2 + 20x + 75
8.
5x2 + 30
35x
−
x−6
x−6
21.
5x
35
−
x2 − 21x + 98 7x − x2
9.
4x2 − 60x
224
+
x−7
x−7
22.
7x
147
+
7x − x2 x2 + 7x − 98
10.
63 − 30x
3x2
−
x−7
7−x
23.
11.
3x2
48 − 30x
−
x−2
2−x
24.
12.
4x2 − 164
20
−
x−6
6−x
25.
13.
81x − 126
9x2
−
x−2
x−2
26.
x2
3x
15
+ 2
− 6x + 5 x − 14x + 45
x2
7x
28
+ 2
− 4x x − 12x + 32
x2
9x
54
− 2
+ 4x − 12 x + 20x + 84
x2
−7x
35
− 2
− 8x + 15 x − 12x + 35
−6x
12
+ 2
+ 2x x + 6x + 8
x2
x2
−9x
36
− 2
− 12x + 32 x − 4x
x2
5x
20
−
− 12x + 32 4x − x2
Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/
Version: Fall 2007
692
27.
28.
29.
Chapter 7
x2
Rational Functions
6x
42
−
− 21x + 98 7x − x2
x2
−2x
4
+ 2
− 3x − 10 x + 11x + 18
x2
−9x
18
− 2
− 6x + 8 x − 2x
37.
Let
f (x) =
x2
11x
+ 12x + 32
and
g(x) =
44
−4x − x2
30.
6x
90
+ 2
2
5x − x
x + 5x − 50
Compute f (x) + g(x) and simplify your
answer.
31.
8x
120
+ 2
2
5x − x
x + 5x − 50
38.
32.
33.
34.
−5x
25
+ 2
+ 5x x + 15x + 50
x2
x2
−5x
30
+ 2
+ x − 30 x + 23x + 102
x2
9x
36
− 2
+ 12x + 32 x + 4x
f (x) =
g(x) =
x2
48
− 18x + 72
Let
f (x) =
4x
−x − x2
and
and
16
g(x) = 2
x + 2x
Compute f (x) − g(x) and simplify your
answer.
g(x) =
x2
Let
Let
−7x
f (x) = 2
x + 8x + 12
f (x) =
x2
Compute f (x) + g(x) and simplify your
answer.
Version: Fall 2007
5x
− x − 12
and
and
42
g(x) = 2
x + 16x + 60
4
+ 3x + 2
Compute f (x) + g(x) and simplify your
answer.
40.
36.
8x
− 6x
Compute f (x) + g(x) and simplify your
answer.
Let
8x
f (x) = 2
x + 6x + 8
x2
and
39.
35.
Let
g(x) =
x2
15
+ 13x + 30
Compute f (x) − g(x) and simplify your
answer.
Section 7.5
Sums and Differences of Rational Functions
7.5 Solutions
1.
Provided x 6= 6,
42
7x2 − 49x + 42
7x2 − 49x
+
=
x−6
x−6
x−6
7(x2 − 7x + 6)
x−6
7(x − 6)(x − 1)
=
x−6
=
= 7(x − 1)
3.
Provided x 6= −3,
27x − 9x2
162
−9x2 + 27x + 162
+
=
x+3
x+3
x+3
−9(x2 − 3x − 18)
x+3
−9(x + 3)(x − 6)
=
x+3
=
= −9(x − 6)
5.
Provided x 6= 4,
4x2 − 8
56
4x2 − 8
56
+
=
−
x−4
4−x
x−4
x−4
=
4x2 − 8 − 56
x−4
=
4x2 − 64
x−4
4(x2 − 16)
x−4
4(x − 4)(x + 4)
=
x−4
=
= 4(x + 4)
Version: Fall 2007
Chapter 7
7.
Rational Functions
Provided x 6= 1,
72x − 63
9x2
72x − 63
9x2
+
=
−
x−1
1−x
x−1
x−1
=
9x2 − 72x + 63
x−1
9(x2 − 8x + 7)
x−1
9(x − 1)(x − 7)
=
x−1
=
= 9(x − 7)
9.
Provided x 6= 7,
4x2 − 60x
224
4x2 − 60x + 224
+
=
x−7
x−7
x−7
4(x2 − 15x + 56)
x−7
4(x − 7)(x − 8)
=
x−7
=
= 4(x − 8)
11.
Provided x 6= 2,
3x2
48 − 30x
3x2
48 − 30x
−
=
+
x−2
2−x
x−2
x−2
=
3x2 − 30x + 48
x−2
3(x2 − 10x + 16)
x−2
3(x − 2)(x − 8)
=
x−2
=
= 3(x − 8)
13.
Provided x 6= 2,
9x2
81x − 126
9x2 − 81x + 126
−
=
x−2
x−2
x−2
9(x2 − 9x + 14)
x−2
9(x − 2)(x − 7)
=
x−2
=
= 9(x − 7)
Version: Fall 2007
Section 7.5
15.
Sums and Differences of Rational Functions
Provided x 6= 3,
15
3x2 − 12
15
3x2 − 12
+
=
−
x−3
3−x
x−3
x−3
=
3x2 − 12 − 15
x−3
=
3x2 − 27
x−3
3(x2 − 9)
x−3
3(x − 3)(x + 3)
=
x−3
=
= 3(x + 3)
17.
3x
15
+ 2
− 6x + 5 x − 14x + 45
3x
15
=
+
(x − 5)(x − 1) (x − 5)(x − 9)
x2
=
3x(x − 9)
15(x − 1)
+
(x − 5)(x − 1)(x − 9) (x − 5)(x − 1)(x − 9)
=
3x(x − 9) + 15(x − 1)
(x − 5)(x − 1)(x − 9)
=
3x2 − 12x − 15
(x − 5)(x − 1)(x − 9)
=
3(x − 5)(x + 1)
(x − 5)(x − 1)(x − 9)
=
3(x + 1)
(x − 1)(x − 9)
Restricted values are 5, 1, and 9.
Version: Fall 2007
Chapter 7
Rational Functions
19.
9x
54
− 2
+ 4x − 12 x + 20x + 84
9x
54
−
=
(x + 6)(x − 2) (x + 6)(x + 14)
x2
=
9x(x + 14)
54(x − 2)
−
(x + 6)(x − 2)(x + 14) (x + 6)(x − 2)(x + 14)
=
9x(x + 14) − 54(x − 2)
(x + 6)(x − 2)(x + 14)
=
9x2 + 72x + 108
(x + 6)(x − 2)(x + 14)
=
9(x + 6)(x + 2)
(x + 6)(x − 2)(x + 14)
=
9(x + 2)
(x − 2)(x + 14)
Restricted values are −6, 2, and −14.
21.
5x
35
−
− 21x + 98 7x − x2
35
5x
+
= 2
x − 21x + 98 x2 − 7x
5x
35
=
+
(x − 7)(x − 14) x(x − 7)
x2
=
5x2
35(x − 14)
+
x(x − 7)(x − 14) x(x − 7)(x − 14)
=
5x2 + 35(x − 14)
x(x − 7)(x − 14)
=
5x2 + 35x − 490
x(x − 7)(x − 14)
=
5(x − 7)(x + 14)
x(x − 7)(x − 14)
=
5(x + 14)
x(x − 14)
Restricted values are 7, 14, and 0.
Version: Fall 2007
Section 7.5
Sums and Differences of Rational Functions
23.
−7x
35
− 2
− 8x + 15 x − 12x + 35
−7x
35
−
=
(x − 5)(x − 3) (x − 5)(x − 7)
x2
=
−7x(x − 7)
35(x − 3)
−
(x − 5)(x − 3)(x − 7) (x − 5)(x − 3)(x − 7)
=
−7x(x − 7) − 35(x − 3)
(x − 5)(x − 3)(x − 7)
=
−7x2 + 14x + 105
(x − 5)(x − 3)(x − 7)
=
−7(x − 5)(x + 3)
(x − 5)(x − 3)(x − 7)
=
−7(x + 3)
(x − 3)(x − 7)
Restricted values are 5, 3, and 7.
25.
−9x
36
− 2
− 12x + 32 x − 4x
36
−9x
=
−
(x − 4)(x − 8) x(x − 4)
x2
=
−9x2
36(x − 8)
−
x(x − 4)(x − 8) x(x − 4)(x − 8)
=
−9x2 − 36(x − 8)
x(x − 4)(x − 8)
=
−9x2 − 36x + 288
x(x − 4)(x − 8)
=
−9(x − 4)(x + 8)
x(x − 4)(x − 8)
=
−9(x + 8)
x(x − 8)
Restricted values are 4, 8, and 0.
Version: Fall 2007
Chapter 7
Rational Functions
27.
6x
42
−
− 21x + 98 7x − x2
6x
42
+
= 2
x − 21x + 98 x2 − 7x
6x
42
=
+
(x − 7)(x − 14) x(x − 7)
x2
=
6x2
42(x − 14)
+
x(x − 7)(x − 14) x(x − 7)(x − 14)
=
6x2 + 42(x − 14)
x(x − 7)(x − 14)
=
6x2 + 42x − 588
x(x − 7)(x − 14)
=
6(x − 7)(x + 14)
x(x − 7)(x − 14)
=
6(x + 14)
x(x − 14)
Restricted values are 7, 14, and 0.
29.
18
−9x
− 2
− 6x + 8 x − 2x
−9x
18
=
−
(x − 2)(x − 4) x(x − 2)
x2
=
−9x2
18(x − 4)
−
x(x − 2)(x − 4) x(x − 2)(x − 4)
=
−9x2 − 18(x − 4)
x(x − 2)(x − 4)
=
−9x2 − 18x + 72
x(x − 2)(x − 4)
=
−9(x − 2)(x + 4)
x(x − 2)(x − 4)
=
−9(x + 4)
x(x − 4)
Restricted values are 2, 4, and 0.
Version: Fall 2007
Section 7.5
Sums and Differences of Rational Functions
31.
8x
120
+ 2
2
5x − x
x + 5x − 50
−8x
120
= 2
+ 2
x − 5x x + 5x − 50
−8x
120
=
+
x(x − 5) (x − 5)(x + 10)
=
−8x(x + 10)
120x
+
x(x − 5)(x + 10) x(x − 5)(x + 10)
=
−8x(x + 10) + 120x
x(x − 5)(x + 10)
=
−8x2 + 40x
x(x − 5)(x + 10)
−8x(x − 5)
x(x − 5)(x + 10)
−8
=
x + 10
=
Restricted values are 5, 0, and −10.
33.
−5x
30
+ 2
+ x − 30 x + 23x + 102
−5x
30
=
+
(x + 6)(x − 5) (x + 6)(x + 17)
x2
=
−5x(x + 17)
30(x − 5)
+
(x + 6)(x − 5)(x + 17) (x + 6)(x − 5)(x + 17)
=
−5x(x + 17) + 30(x − 5)
(x + 6)(x − 5)(x + 17)
=
−5x2 − 55x − 150
(x + 6)(x − 5)(x + 17)
=
−5(x + 6)(x + 5)
(x + 6)(x − 5)(x + 17)
=
−5(x + 5)
(x − 5)(x + 17)
Restricted values are −6, 5, and −17.
Version: Fall 2007
Chapter 7
Rational Functions
35.
8x
16
− 2
+ 6x + 8 x + 2x
8x
16
−
=
(x + 2)(x + 4) x(x + 2)
f (x) − g(x) =
x2
=
8x2
16(x + 4)
−
x(x + 2)(x + 4) x(x + 2)(x + 4)
=
8x2 − 16(x + 4)
x(x + 2)(x + 4)
=
8x2 − 16x − 64
x(x + 2)(x + 4)
=
8(x + 2)(x − 4)
x(x + 2)(x + 4)
=
8(x − 4)
x(x + 4)
Restricted values are −2, −4, and 0.
37.
11x
44
+
x2 + 12x + 32 −4x − x2
11x
44
= 2
− 2
x + 12x + 32 x + 4x
11x
44
=
−
(x + 4)(x + 8) x(x + 4)
f (x) + g(x) =
=
44(x + 8)
11x2
−
x(x + 4)(x + 8) x(x + 4)(x + 8)
=
11x2 − 44(x + 8)
x(x + 4)(x + 8)
=
11x2 − 44x − 352
x(x + 4)(x + 8)
=
11(x + 4)(x − 8)
x(x + 4)(x + 8)
=
11(x − 8)
x(x + 8)
Restricted values are −4, −8, and 0.
Version: Fall 2007
Section 7.5
Sums and Differences of Rational Functions
39.
4x
4
+ 2
2
−x − x
x + 3x + 2
−4x
4
+
= 2
x + x x2 + 3x + 2
−4x
4
=
+
x(x + 1) (x + 1)(x + 2)
f (x) + g(x) =
=
−4x(x + 2)
4x
+
x(x + 1)(x + 2) x(x + 1)(x + 2)
=
−4x(x + 2) + 4x
x(x + 1)(x + 2)
=
−4x2 − 4x
x(x + 1)(x + 2)
=
−4x(x + 1)
x(x + 1)(x + 2)
=
−4x
x(x + 2)
=
−4
x+2
Restricted values are −1, 0, and −2.
Version: Fall 2007