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C H A P T E R 6Rational Expressions, Equations, and Functions
210 Chapter 6 Rational Expressions, Equations, and Functions
42.
�4
3 � x � yx2y2 �4
3x3y3
�5 � 5 � x2 � y � 4 � 4 � x2 � y
3 � 4 � 5 � x2 � x � y � y � 5 � x2 � x2 � y � y2
25x2y60x3y2 �
5x 4y3
16x2y�
25x2y60x3y2 �
16x2y5x 4y3
44.
��x � 2��x2 � 9�
�x � 3� , x � ±2
x2 � 9
5�x � 2� �x � 3
5�x2 � 4� �x2 � 9
5�x � 2� �5�x � 2��x � 2�
x � 346.
�y�x � y�
�x � 4��x � y�, x � 0, y � 0
��x � y��x � y� � 2 � x � y
2 � x�x � 4��x � y��x � y�
x2 � y2
2x2 � 8x�
�x � y�2
2xy�
x2 � y2
2x2 � 8x�
2xy�x � y�2
52.
�x � 3
x�2x � 1�, x � ±3
��x � 3��2x � 1�
x2�x � 3� �x�x � 3��2x � 1�2
�x2 � 6x � 9x2 �
2x � 1x2 � 9 � �
4x2 � 4x � 1x2 � 3x
� ��x � 3�2
x2 �2x � 1
�x � 3��x � 3�� ��2x � 1�2
x�x � 3�
48.
�1
x � 5, x � �3, �2
x � 3
x2 � 7x � 10�
x2 � 6x � 9x2 � 5x � 6
�x � 3
�x � 5��x � 2� ��x � 3��x � 2��x � 3��x � 3�
50.
��y � 2��y � 4��y � 3��y � 2�, y � �4, �7
y2 � 5y � 14
y2 � 10y � 21�
y2 � 5y � 6y2 � 7y � 12
��y � 7��y � 2��y � 7��y � 3� �
�y � 4��y � 3��y � 3��y � 2�
56.
�5�t � 5�
4t 3 , t � �10, 10
��t � 10��t � 10�t�t � 10��t � 5�5t
4t 2 � t 3�t � 10��t � 10�2
t 2 � 100
4t 2 �t 3 � 5t 2 � 50t
t 4 � 10t 3 ��t � 10�2
5t�
�t � 10��t � 10�4t 2 �
t�t 2 � 5t � 50�t3�t � 10� �
5t�t � 10�2
54.
�u�u � 3��3u � 4��u � 1�
3u2 � 12u � 4, u � 3, 0
��3u � 4��u � 1�
u2 �u3�u � 3�
3u2 � 12u � 4
3u2 � u � 4
u2 �3u2 � 12u � 4
u4 � 3u3 ��3u � 4��u � 1�
u2 �3u2 � 12u � 4
u3�u � 3�
58. xn�1 � 8x
x2n � 2xn � 1�
x2n � 4xn � 5x
� xn �x�xn � 8�
�xn � 1��xn � 1� ��xn � 5��xn � 1�
x�
1xn �
�xn � 8��xn � 5�xn�xn � 1�
Section 6.3 Adding and Subtracting Rational Expressions 211
68. (a)
(b)
(c)4 seconds1 gallon
� 130 gallons � 520 seconds or 52060
�263
minutes
4 seconds1 gallon
� x gallons � 4x seconds or x
15 minutes
15 gallons1 minute
�15 gallons60 seconds
�1 gallon
4 seconds, t � 4 seconds or
115
minute
72.
The value of the first row gets larger and closer to 1 as the value of increases (because as becomes larger, the value of 10becomes much smaller in comparison). The value of the second row gets smaller and closer to 1 as the value of increases(because as becomes larger, the value of 50 becomes much smaller in comparison). The value of the third row is in betweenthe values of the other two rows and gets smaller and closer to 1 as the value of increases.x
Section 6.5 Dividing Polynomials and Synthetic Division 225
82.
5xn � 5
0x2n � 5xn � 5
x3n � x2n
xn � 1 ) x3n � x2n � 5xn � 5
x2n � 5, xn � 1
84.
� x4 � 4x3 � 3x2 � 4x � 4
� x4 � x3 � 4x � 3x3 � 3x2 � 12 � 8
� �x � 3��x3 � x2 � 4� � 8
Dividend � Divisor � Quotient � Remainder
86.
The polynomial values equal the remainders.
� 9
� 16 � 4 � 4 � 1
f �2� � 2�2�3 � �2�2 � 2�2� � 1
� 0
� 2 � 1 � 2 � 1
f �1� � 2�1�3 � �1�2 � 2�1� � 1
� 0
� 14 �
14 � 1 � 1
f �12� � 2�1
2�3� �1
2�2� 2�1
2� � 1
� 1
� 0 � 0 � 0 � 1
f �0� � 2�0�3 � �0�2 � 2�0� � 1
� 0
� �2 � 1 � 2 � 1
f ��1� � 2��1�3 � ��1�2 � 2��1� � 1
� �15
� �16 � 4 � 4 � 1
f ��2� � 2��2�3 � ��2�2 � 2��2� � 1
2 2
2
�14
3
�26
4
18
9
1 2
2
�12
1
�21
�1
1�1
0
12 2
2
�11
0
�20
�2
1�1
0
0 2
2
�10
�1
�20
�2
10
1
�1 2
2
�1�2
�3
�23
1
1�1
0
�2 2
2
�1�4
�5
�210
8
1�16
�15
Divisors, Remainder
0 0
0 1 1
0 0
1 0 0
2 9 9x � 2
x � 1
x �12
12
x
x � 1�1
�15x � 2�15�2
�x � k�f �k�k
80. Keystrokes:
2 1
1 3 1
Thus,x2 � 2x � 1
� x � 1 �3
x � 1.
�1 1
1
0�1
�1
21
3
y2
y1−15
−12
15
8
Y� � X,T, � x2 � � � X,T, � �
�
ENTER
X,T, � � � � � X,T, � �
�
GRAPH
�
226 Chapter 6 Rational Expressions, Equations, and Functions
88.
Area of first floor � x2 � 50x � 400 (square feet)
�5 1
1
55 �5
50
650 �250
400
2000�2000
0
Area of first floor �VolumeHeight
�x3 � 55x2 � 650x � 2000
x � 5
Volume � Area of first floor � Height
Volume � Length � Width � Height 90.
� h2 � 2h
� h�h � 2�
�h2�h � 2��h � 1�
h�h � 1�
�h2�h2 � 3h � 2�
h�h � 1�
�h4 � 3h3 � 2h2
h�h � 1�
Length �Volume
Width � Height
Volume � Length � Width � Height
92.
Divisor:
Dividend:
Quotient:
Remainder: 5
x � 1
x2 � 4
x � 1
x2 � 4x � 1
� x � 1 �5
x � 1
94. Check polynomial division by multiplication. Using Exercise 92 as an example:
� x2 � 4
� x2 � 1 � 5
�x � 1��x � 1 �5
x � 1� � �x � 1��x � 1� � �x � 1� 5x � 1
96. For synthetic division, the divisor must be of the form x � k.
98.
Keystrokes:
3 5 2 8
2
The function has -intercepts and It has only two -intercepts because the numerator factors and it is possible to divide out the same factor in the numerator and denominator, leaving a second-degree polynomial having two factors. Thefunction appears to be equivalent to The difference is at x � 2.f �x� � �x � 4��x � 1�.
x�4, 0�.��1, 0�x
−7 11
−8
4
f �x� �x3 � 5x2 � 2x � 8
x � 2
Y� � X,T, � �
�
�X,T, � X,T, ��
�
�
�
� X,T, � GRAPH
x2>
Section 6.6 Solving Rational Equations 227
Section 6.6 Solving Rational Equations
2. (a)
Not a solution
210
is undefined.
0 �?
4 �210
x � 0 (b)
Solution
�3 � �3
�3 �?
4 � ��7�
�3 �?
4 �21�3
x � �3 (c)
Solution
7 � 7
7 �?
4 � 3
7 �?
4 �217
x � 7 (d)
Not a solution
�1 � �17
�1 �?
4 � 21
�1 �?
4 �21�1
x � �1
4. (a)
Solution
2 � 2
5 � 3 �?
2
5 �1 13
�?
2
5 �1
103 �
93
�?
2
5 �1
103 � 3
�?
2
x �103
(b)
Not a solution
5310
� 2
5010
�3
10�?
2
5 � ��3
10� �?
2
5 �1
�103
�?
2
5 �1
�13 �
93
�?
2
5 �1
�13 � 3
�?
2
x � �13
(c)
Not a solution
163
� 2
153
�13
�?
2
5 � ��13� �
?2
5 �1
0 � 3�?
2
x � 0 (d)
Not a solution
112
� 2
102
�12
�?
2
5 �1
�2�?
2
5 �1
1 � 3�?
2
x � 1
6.
y � �60
y � 56 � �4
8�y8
� 7� � ��12�8
y8
� 7 � �12
Check:
�12
� �12
�15
2�
142
�?
�12
�60
8� 7 �
?�
12
12.
�409
� x
40 � �9x
4x � 40 � �5x
4x � 20 � 60 � �5x
4�x � 5� � 60 � �5x
20�x � 55
� 3� � ��x4�20
x � 5
5� 3 � �
x4
Check:
109
�109
�179
�279
�? 10
9
15��
859 � � 3 �
? 109
�
409 �
459
5� 3 �
? 409
�14
�
409 � 5
5� 3 �
?�
�409
4
8.
a � 5
�3a � �15
2a � 5a � 15
2a � 5�a � 3�
a
5�
a � 3
2Check:
1 � 1
1 �? 2
2
55
�? 5 � 3
2
10.
x � 3
3x � 2x � 3
12�x4
�x6� � �1
4�12
x4
�x6
�14
Check:
14
�14
34
�24
�? 1
4
34
�12
�? 1
4
34
�36
�? 1
4
228 Chapter 6 Rational Expressions, Equations, and Fractions
14.
�9
20� x
�9 � 20x
8x � 4 � 5 � 28x
2�4x � 2� � 5 � 28x
14�4x � 27
�514� � �2x�14
4x � 2
7�
514
� 2x Check:
�910
��910
�6370
�? �9
10
�3870
�2570
�? �9
10
�1935
�514
�? �9
10
�19
5�
17
�5
14�? �9
10
�95 �
105
7�
514
�? �9
10
4��920 � � 2
7�
514
�?
2��920 �
16.
x �15
x � 1
5x � 1 � 0 x � 1 � 0
�5x � 1��x � 1� � 0
5x2 � 6x � 1 � 0
5x2 � 6x � �1
10�x2
2�
3x5 � � ��1
10 �10
x2
2�
3x5
��110
Check:
�110
��110
�5
50�? �1
10
150
�6
50�? �1
10
150
�3
25�? �1
10
1252
�
355
�? �1
10
�1
5�2
2�
3�15�5
�? �1
10
Check:
�1
10�
�110
510
�6
10�? �1
10
12
�35
�? �1
10
12
2�
3�1�5
�? �1
10
20.
u � 10
9u � 90
5u � 10 � 4u � 10 � 90
5�u � 2� � 2�2u � 5� � 90
30�u � 26
�2u � 5
15 � � �3�30
u � 2
6�
2u � 515
� 3 Check:
3 � 3
93
�?
3
43
�53
�?
3
86
�2515
�?
3
10 � 2
6�
2�10� � 515
�?
3
18.
z � �36
�z � 36
�z � 9 � 27
2z � 8 � 3z � 1 � 27
2�z � 4� � �3z � 1� � 27
18�z � 49
�3z � 1
18 � � �32�18
z � 4
9�
3z � 118
�32
Check:
32
�32
2718
�? 3
2
�8018
�10718
�? 3
2
�40
9�
�10718
�? 3
2
�36 � 4
9�
3��36� � 118
�? 3
2
Section 6.6 Solving Rational Equations 229
24.
�45
� u
�4 � 5u
16 � 5u � 20
8�2� � 5�u � 4�
8�u � 4�� 2u � 4� � �5
8�8�u � 4�
2
u � 4�
58
Check:
58
�58
1016
�? 5
8
2
165
�? 5
8
2
�45 �
205
�? 5
8
2
�45 � 4
�? 5
8
26.
3 � b
6 � 2b
b�6b� � �2�b
6b
� 2
6b
� 22 � 24 Check:
24 � 24
2 � 22 �?
24
63
� 22 �?
24 28.
x �127
x �6035
35x � 60
35x � 18 � 42
21x�53� � � 6
7x�
2x�21x
53
�67x
�2x
Check:
53
�53
53
�? 20
12
53
�? 6
12�
1412
53
�? 6
7�127 � �
2127
30.
t � 130
7t � 6t � 130
7t � 142 � 6t � 12
7t � 14 � 128 � 6t � 12
7�t � 2� � 16�8� � 6�t � 2�
8�t � 2��78
�16
t � 2� � �34�8�t � 2�
78
�16
t � 2�
34
Check:
34
�34
68
�? 3
4
78
�18
�? 3
4
78
�16
128�? 3
4
78
�16
130 � 2�? 3
4
22.
x ��21
2
�2x � 21
�4x � 9 � 12 � 2x
2x � 7 � 6x � 2 � 12 � 2x
2x � 7 � 2�3x � 1� � 2�6 � x�
10�2x � 710
�3x � 1
5 � � �6 � x5 �10
2x � 7
10�
3x � 15
�6 � x
5Check:
3310
�3310
�2810
�6110
�? 33
10
�2810
�
�612
5�?
332
5
�21 � 7
10�
�632 �
22
5�?
122 �
212
5
2��21
2 � � 710
�3��21
2 � � 15
�? 6 � ��21
2 �5
230 Chapter 6 Rational Expressions, Equations, and Fractions
34.
x � 5
x �1750350
350x � 1750
350x � 1500 � 250
500x � 1500 � 150x � 250
500�x � 3� � 50�3x � 5�
�x � 3��3x � 5�� 5003x � 5� � � 50
x � 3��x � 3��3x � 5�
500
3x � 5�
50x � 3
Check:
25 � 25
50020
�?
25
500
15 � 5�? 50
2
500
3�5� � 5�? 50
5 � 3
36.
�25 � x
�75 � 3x
25 � 3x � 100
17x � 25 � 20x � 100
12x � 5x � 25 � 20x � 100
12x � 5�x � 5� � 20�x � 5�
x�x � 5�� 12x � 5
�5x� � �20
x �x�x � 5�
12
x � 5�
5x
�20x
Check:
�45
� �45
3
�5�
1�5
�?
�45
12
�20�
5�25
�? 20
�25
12
�25 � 5�
5�25
�? 20
�25
32.
x �45
x �2025
25x � 20
25x � 40 � 60
40x � 40 � 15x � 60
40�x � 1� � 15�x � 4�
4 � 10�x � 1� � 15�x � 4�
4�x � 1��x � 4�� 10x � 4� � � 15
4�x � 1��4�x � 1��x � 4�
10
x � 4�
154�x � 1� Check:
2512
�2512
5024
�? 75
36
5024
�? 15
365
10245
�? 15
4�95�
10
45 �
205
�? 15
4�45 �
55�
10
45 � 4
�? 15
4�45 � 1�
Section 6.6 Solving Rational Equations 231
42.
x � 8 x � �6
0 � �x � 8��x � 6�
0 � x2 � 2x � 48
48 � x2 � 2x
x�48x � � �x � 2�x
48x
� x � 2 Check:
Check:
�8 � �8
48�6
�?
�6 � 2
6 � 6
488
�?
8 � 2 44.
x � �3
x � 8
�x � 8��x � 3� � 0
x2 � 5x � 24 � 0
x2 � 24 � 5x
x�x �24x � � �5�x
x �24x
� 5 Check:
Check:
5 � 5
�3 � 8 � 5
�3 �24�3
�?
5
5 � 5
8 � 3 � 5
8 �248
�?
5
46.
x � 7 x � �6
x � 7 � 0 x � 6 � 0
0 � �x � 7��x � 6�
0 � x2 � x � 42
x � 42 � x2
x�x � 42x � � �x�x
x � 42
x� x Check: Check:
�6 � �6 7 � 7
36�6
�?
�6 497
�?
7
�6 � 42
�6�?
�6 7 � 42
7�?
7
40.
u � 10 u � �10
u � 10 � 0 u � 10 � 0
0 � �u � 10��u � 10�
0 � u2 � 100
100 � u2
5u�20u � � �u
5�5u
20u
�u5
Check:
Check:
�2 � �2
20�10
�? �10
5
2 � 2
2010
�? 10
5
38.
z � 8 z � �8
z � 8 � 0 z � 8 � 0
�z � 8��z � 8� � 0
z2 � 64 � 0
z2 � 64
4�14� � �16
z2 �4
14
�16z2 Check:
Check:
14
�14
14
�? 16
64
14
�? 16
��8�2
14
�14
14
�? 16
64
14
�? 16
�8�2
232 Chapter 6 Rational Expressions, Equations, and Fractions
236 Chapter 6 Rational Expressions, Equations, and Fractions
68. (a) intercept:
(b)
(a) and (b) �0, 0�
0 � x
0 � 2x
�x � 4��0� � � 2xx � 4��x � 4�
0 �2x
x � 4
�0, 0�x-
72. (a) Keystrokes:
1 3 4
intercept:
(b)
�2, 0�
x � 2
2x � 4
0 � �2x � 4
0 � x � 4 � 3x
x�x � 4��0� � �1x
�3
x � 4�x�x � 4�
0 �1x
�3
x � 4
�2, 0�x-
−12
−6
6
6
70. (a) intercepts:
(b)
(a) and (b) ��1, 0�, �2, 0�
x � �1 x � 2,
x � 1 � 0 x � 2 � 0
0 � �x � 2��x � 1�
0 � x2 � x � 2
0 � x2 � 2 � x
x�0� � �x �2x
� 1�x
0 � x �2x
� 1
��1, 0� and �2, 0�x-
Y� � � � � GRAPHX,T,� X,T,� �
66.
x � 2 x � �32
x � 2 � 0 2x � 3 � 0
�2x � 3��x � 2� � 0
2x2 � x � 6 � 0
2x2 � 2x � 3x � 6
3�x � 1�2x3
� 3�x � 1��x � 2��x � 1�
2x3
��x � 2��x � 1�
2x3
��1 �
2x�x
�1 �1x�x
2x3
�
1 �2x
1 �1x
Check:
�1 � �1
�1 �?
�13
13
�33
�?
1 �43
1 �23
2��
32�
3�?
1 �2
��3�2�
1 �1
��3�2�
Check:
43
�43
43
�? 2
32
2�2�
3�?
1 �22
1 �12
�
Section 6.6 Solving Rational Equations 237
76. (a) Keystrokes:
4
intercepts: and
(b)
��2, 0��2, 0�
x � �2 x � 2
x � 2 � 0 x � 2 � 0
0 � �x � 2��x � 2�
0 � x2 � 4
x�0� � �x2 � 4x �x
0 �x2 � 4
x
�2, 0���2, 0�x- −15
−10
15
10
� � �
�
GRAPH� x2X,T,� X,T,�
78. expression
3�5�3�x � 3� �
5�x � 3�3�x � 3� �
3�3��x � 3�3�x � 3� �
15 � 5x � 15 � 9x � 273�x � 3� �
14x � 573�x � 3�
5x � 3
�53
� 3 →
80.
34
� x
3 � 4x
15 � 5x � 15 � 9x � 27
15 � 5�x � 3� � 9�x � 3�
3�x � 3�� 5x � 3
�53� � �3�3�x � 3�
5
x � 3�
53
� 3 → equation
74. (a) Keystrokes:
20 2 3 1
intercept:
(b)
��2, 0�
x � �2
20x � �40
0 � �20x � 40
0 � 40x � 40 � 60x
0 � 40�x � 1� � 60x
x�x � 1��0� � �20�2x
�3
x � 1��x�x � 1�
0 � 20�2x
�3
x � 1���2, 0�x-
−3
−400
4
400Y� � � � � �
GRAPH� �
�X,T,� X,T,�
238 Chapter 6 Rational Expressions, Equations, and Fractions
84.
Labels:
Choose the positive value of The two average speeds are 50 miles per hour and 60 miles per hour.
r.
r � 10 � speed second part
r � speed first part
86.
Labels:
Thus, the speed of the jet is approximately 246 miles per hour and the speed of the commuter plane is approximately miles per hour.246 � 150 96
x � 150 � rate of commuter plane
x � rate of jet
Distance traveled by commuter planeRate of commuter plane
VerbalModel:
�Distance traveled by jet
Rate of jet
Equation:
x 246.43
�700x � �172,500
450x � 1150x � 172,500
450x � 1150�x � 150�
x�x � 150�� 450x � 150� � �1150
x �x�x � 150�
450
x � 150�
1150x
Equation:
r � 50 r � �8
r � 50 � 0 3r � 24 � 0
�r � 50��3r � 24� � 0
3r2 � 126r � 1200 � 0
6r2 � 252r � 2400 � 0
312r � 2400 � 6r2 � 60r
�r � 10�240 � r�72� � 6r2 � 60r
r�r � 10��240r � � r�r � 10�� 72
r � 10� � r�r � 10�6
240
r�
72r � 10
� 6
Distance first partSpeed first part
Distance second partSpeed second part
Totaltime
VerbalModel:
82.
Labels:
Equation:
x �18
x � 12 8x � 1
x � 12 � 0 8x � 1 � 0
�8x � 1��x � 12� � 0
8x2 � 97x � 12 � 0
8x2 � 12 � 97x
4x�2x �3x� � �97
4 �4x
2x �3x
�974
1x
� reciprocal of number
x � a number
Verbal Model: � �974
Twice a number 3 times the reciprocal
� �
Section 6.7 Applications and Variation 239
2. C � kr 4. s � kt3 6. V � k 3�x 8. S �kv2
10. P �k
�1 � r12. V � khr2 14. F �
km1m2
r2
88. Solve a rational equation by multiplying both sides of theequation by the lowest common denominator. Then solve theresulting equation, checking for any extraneous solutions.
90. Graph the rational equation and approximate any -inter-cepts of the graph.
x
Section 6.7 Applications and Variation
16. The area of a rectangle variesjointly as the length and the width.
18. The volume of a sphere variesdirectly as the cube of the radius.
20. The height of a cylinder variesdirectly as the volume and inverselyas the square of the radius.
22.
h �73r
73 � k
2812 � k
28 � k�12�
h � kr 24.
M � 1.5n3
1.5 � k
0.0120.008
� k
0.012 � k�0.008�
0.012 � k�0.2�3
M � kn3 26.
q �75P
75 � k
150 � 2k
32
�k
50
q �kP
28.
u �10v2
10 � k
40 � 4k
40 �k14
40 �k
�12�2
u �kv2 30.
V �13
hb2
13
� k
288864
� k
288 � 864k
288 � k�6��144�
288 � k�6��12�2
V � khb2 32.
z �135x�y
135 � k
6480 � 48k
720 �48k9
720 �k�48��81
z �kx�y
240 Chapter 6 Rational Expressions, Equations, and Functions