333202_0A03.qxd
12/6/05
2:14 PM
Page A33
Appendix A.3
In Exercises 159–192, completely factor the expression.
159. 6x 54
2
160. 12x 2 48
199.
162. x 3 9x
163. x 2 2x 1
164. 16 6x 167.
4x 168.
2y 3
7y 2
2x 3
xc
200. 2x 2 9x c
and the total revenue R (in dollars) is
15y
R 95x.
169. 9x 2 10x 1
(a) Find the profit P in terms of x.
170. 13x 6 5x 2
172.
198. 3x 2 10x c
C 73x 25,000
166. 9x 2 6x 1
2x 2
3x 2
201. Cost, Revenue, and Profit An electronics manufacturer
can produce and sell x radios per week. The total cost C
(in dollars) of producing x radios is
x2
165. 1 4x 4x 2
171.
In Exercises 197–200, find two integer values of c such that
the trinomial can be factored. (There are many correct
answers.)
197. 2x 2 5x c
161. x 3 4x 2
(b) Find the profit obtained by selling 5000 radios per
week.
1 2
2
81 x 9 x 8
1 2
1
1
8 x 96 x 16
202. Cost, Revenue, and Profit An artisan can produce and
sell x hats per month. The total cost C (in dollars) of
producing x hats is
173. 3x 3 x 2 15x 5
174. 5 x 5x 2 x 3
175. x 4 4x 3 x 2 4x
C 460 12x
176. 3u and the total revenue R (in dollars) is
6
2u2
u3
1
3
177. 4 x3 3x 2 4 x 9
R 36x.
1
178. 5 x3 x2 x 5
(a) Find the profit P in terms of x.
179. t 1 2 49
(b) Find the profit obtained by selling 42 hats per month.
180. x 2 1 2 4x 2
181. x2
182.
2t 3
8 2
203. Compound Interest After 2 years, an investment of
$500 compounded annually at an interest rate r will yield
an amount of 5001 r 2.
36x 2
16
183. 5x 3 40
(a) Write this polynomial in standard form.
184. 4x2x 1 2x 1 2
(b) Use a calculator to evaluate the polynomial for the
values of r shown in the table.
185. 53 4x 2 83 4x5x 1
186. 2x 1x 3 2 3x 1 2x 3
212%
187. 73x 2 21 x 2 3x 21 x3
r
188. 7x2
5001 r
x2
A33
Polynomials and Factoring
12x x2
1 7
2
3%
4%
412%
5%
2
189. 3x 22x 14 x 2 34x 1 3
190. 2xx 5 4 x 24x 5 3
(c) What conclusion can you make from the table?
191. 5x 1 6x 3x 2 33x 2 3x 1
6
192.
x2
2
4
5
3
2
6
5
x2 14 x 2 15
In Exercises 193–196, find all values of b for which the
trinomial can be factored.
193. x 2 bx 15
194. x 2 bx 50
195. x 2 bx 12
196. x 2 bx 24
204. Compound Interest After 3 years, an investment of
$1200 compounded annually at an interest rate r will
yield an amount of 12001 r3.
(a) Write this polynomial in standard form.
(b) Use a calculator to evaluate the polynomial for the
values of r shown in the table.
r
2%
3%
312%
4%
412%
12001 r3
(c) What conclusion can you make from the table?
333202_0A04.qxd
12/6/05
A44
2:16 PM
Appendix A
Page A44
Review of Fundamental Concepts of Algebra
In Exercises 43–52, perform the addition or subtraction and
simplify.
43.
5
x
x1 x1
44.
2x 1 1 x
x3
x3
61. x 5 2x2
62. x5 5x3
63. x 2x 2 15 x 2 14
3
46.
5
x1
5
45. 6 x3
In Exercises 61– 66, factor the expression by removing the
common factor with the smaller exponent.
64. 2xx 53 4x 2x 54
3
5
47.
x2 2x
65. 2x 2x 112 5x 112
66. 4x 32x 132 2x2x 112
2x
5
48.
x5 5x
In Exercises 67 and 68, simplify the expression.
x
1
2
49. 2
x x 2 x 5x 6
67.
3x13 x23
3x23
2
10
x 2 x 2 x 2 2x 8
68.
x 31 x 212 2x1 x 212
x4
50.
51. 2
1
1
2
x
x 1 x3 x
In Exercises 69–72, simplify the difference quotient.
2
2
1
2
52.
x1 x1 x 1
69.
In Exercises 53 and 54, describe the error.
Error Analysis
53.
x 4 3x 8 x 4 3x 8
x2
x2
x2
71.
2x 4 2x 2
2
x2
x2
73.
x6 x x 2 2 8
x 2x 2
6x x 2 x 2 4 8
x 2x 2
6x 2
6
x2x 2 x 2
(x h)
1
70.
h
x h1 4 x 1 4
h
72.
2
1
x2
h
x
xh
xh1 x1
h
In Exercises 73–76, simplify the difference quotient by
rationalizing the numerator.
6x
x2
8
2
54.
xx 2
x2
x x 2
x 1 h 1x 74.
75.
76.
x 2 x
2
z 3 z
3
x h 1 x 1
h
x h 2 x 2
h
In Exercises 55– 60, simplify the complex fraction.
x
1
2
55.
x 2
2
x
2
3
59.
1
2x
x
x 1 57.
x
x 1 x x 4
x
4
4
x
x2 1
x
58.
x 12
x
t2
t 2 1
t 2 1
60.
t2
56.
Probability In Exercises 77 and 78, consider an experiment in which a marble is tossed into a box whose base is
shown in the figure. The probability that the marble will
come to rest in the shaded portion of the box is equal to
the ratio of the shaded area to the total area of the figure.
Find the probability.
77.
78.
x
2
x
2x + 1
x+4
x
x
x+2
4
x
(x + 2)
333202_App_AN.qxd
A322
46.
48.
50.
52.
55.
58.
60.
62.
64.
66.
68.
69.
70.
72.
74.
77.
79.
82.
85.
88.
91.
93.
95.
98.
100.
102.
104.
106.
108.
110.
112.
114.
117.
120.
123.
125.
126.
128.
129.
130.
132.
134.
136.
138.
140.
142.
144.
146.
148.
150.
151.
12/9/05
10:20 AM
Page A322
Answers to All Exercises and Tests
47. x 2 7x 12
74 y 2 8y
2
49. 6x 2 7x 5
x 5x 50
2
51. x 4 x 2 1
28x 29x 6
4
2
53. x 2 100
54. 4x 2 9
x 13x 4
2
2
2
2
56. 4x 9y
57. 4x 2 12x 9
x 4y
2
2
59. 4x 20xy 25y 2
16x 40x 25
2
61. x 3 3x 2 3x 1
64x 80x 25
3
2
63. 8x 3 12x 2y 6xy 2 y 3
x 6x 12x 8
3
2
2
65. 16x 6 24x 3 9
27x 54x y 36xy 8y 3
2
2
2
67. m n 6m 9
64x 48x 9
x 2 2xy y 2 1
x2 2xy y 2 6x 6y 9
71. 4r 4 25
x 2 2xy y 2 2x 2y 1
1 2
6
4
73. 4 x 3x 9
9a 16b
4 2
20
1
75. 91 x2 4
76. 4x2 25
t
t
25
9
3
2
2
78. 2.25y 9y 9
1.44x 7.2x 9
80. 6.25y 2 9
81. 2x2 2x
2.25x2 16
2
4
83. u 16
84. x 4 y 4
2x 8x 6
2
86. 25 x
87. x 25 x 5
xy
89. 3x 2
90. 5 y 6
x2 2x3 3
92. 2x2x 2 3x 6
2xx 2 3
94. x 23x 4
x 1x 6
96. 3x3x 1
97. 12 x 8
x 3x 1
1
1
2
99. 2 xx 4x 10
3 y 15
1
3 15y 6
101. 23 x 6x 3
y
y
3
2
103. x 9x 9
5 y 12y 5
105. 24y 34y 3
x 7x 7
107. 4x 13 4x 13 43y 13y 1
2
2
5 y 85 y 8 109. x 1x 3
111. 3u 2v3u 2v
z z 10
113. x 2 2
5x 4y5x 4y
115. 2t 1 2
116. 3x 2 2
x 5 2
118. 6y 92
119. 3u 4v2
5y 1 2
2
2
121. x 23 122. z 12 2x y2
124. x 3x 2 3x 9
x 2x 2 2x 4
y 4 y 2 4y 16
127. 2t 14t 2 2t 1
z 5z 2 5z 25
3x 29x 2 6x 4
u 3vu2 3uv 9v2
131. x 2x 1
4x y16x2 4xy y 2
133. s 3s 2
x 2x 3
135. y 5 y 4
t 2t 3
137. x 20x 10
z 8z 3
139. 3x 2x 1
x 6x 7
141. 5x 1x 5
2x 1x 1
143. 3z 23z 1
3x 14x 1
145. x 1x 2 2
5u 2u 3
147. 2x 1x 2 3
x 5x 2 5
149. 3 x2 x 3
x 25x 2 3
x2 2x 1x2 x 1
152. 4x 3 32x2 3
3x2 12x 1
153.
155.
157.
159.
161.
164.
166.
168.
170.
172.
174.
176.
178.
180.
181.
182.
183.
185.
187.
188.
190.
191.
192.
193.
195.
196.
197.
198.
199.
200.
201.
202.
203.
154. x 32x 3
x 23x 4
156. 2x 33x 5
2x 13x 2
158. x 112x 1
3x 15x 2
160. 12x 2x 2
6x 3x 3
162. xx 3x 3
163. x 1 2
x 2x 4
2
165. 1 2x
8 x2 x
167. 2xx 1x 2
3x 13x 1
169. 9x 1x 1
y2y 3 y 5
1
171. 81
5x 3)x 2
x 36x 18
1
173. 3x 1x 2 5
96 4x 33x 2
175. xx 4x 2 1
5 x1 x 2
2
177. 14 x2 3x 12
u 23 u 1 2
179. t 6t 8
5 x 5x 5
x 1 2 x 1 2
x 2x 4x 2x 4
2t 2t 2 2t 4
184. 2x 16x 1
5x 2x 2 2x 4
186. x 1x 3x 9
3 4x23 60x
51 x 23x 24x 3
189. x 2 2x 1 37x 5
7x 2 13x 2 1
3
2x x 5 x 5
3x6 143x 2233x6 20x5 3
x2
1
x 2 1 4
2
194. 51, 51, 15, 15, 27, 27
14, 14, 2, 2
11, 11, 4, 4, 1, 1
25, 25, 14, 14, 11, 11, 10, 10
Two possible answers: 2, 12
Two possible answers: 3, 8
Two possible answers: 2, 4
Two possible answers: 9, 7
(a) P 22x 25,000
(b) $85,000
(a) P 24x 460
(b) $548
(a) 500r 2 1000r 500
(b)
r
3%
4%
212%
5001 r 2
$525.31
$530.45
r
412%
5%
5001 r 2
$546.01
$551.25
$540.80
(c) The amount increases with increasing r.
204. (a) 1200r 3 3600r 2 3600r 1200
(b)
r
2%
3%
312%
12001 r 3
$1273.45 $1311.27 $1330.46
r
4%
12001 r 3
$1349.84 $1369.40
412%
(c) The amount increases with increasing r.
333202_App_AN.qxd
A324
12/9/05
10:20 AM
Page A324
Answers to All Exercises and Tests
1
, x ±3
27. z 2
x1
y y 3
28. 2
, y 1
y y1
29. x
0
1
2
26.
62.
65.
3
4
5
6
67.
x 2x 3
x3
1
2
3
Undef.
5
6
7
69.
x1
1
2
3
4
5
6
7
71.
2
The expressions are equivalent except at x 3.
30.
0
1
2
3
4
5
6
x3
x2 x 6
1
2
1
3
1
4
Undef.
1
6
1
7
1
8
74.
1
2
1
3
1
4
1
5
1
6
1
7
1
8
76.
1
x2
31.
32.
33.
35.
37.
39.
41.
43.
47.
49.
51.
53.
54.
55.
57.
59.
72.
x
The expressions are equivalent except at x 3.
The expression cannot be simplified.
The polynomial, x2 25, can not be factored.
x5
34.
, r0
, x 5
4
42x 3
1
x 13
36. , x1
, x3
5x 2
5x 2
r1
8
38. , y 3, 4
, r1
r
5
t3
xy
40.
, t 2
, x 2y
t 3t 2
xx y 2
x 6x 1
1
42. , x ± 7
, x6
x2
3
x5
x
6x 13
8 5x
44.
45.
46.
x1
x3
x3
x1
2
2x 5
48.
x2
x5
62x 3
x2 3
50.
x 1x 2x 3
x 2x 1x 4
2x
4x 1
52.
, x0
x2 1
x 1x 1
The error was incorrect subtraction in the numerator.
The error was an incorrect expansion of x 22 in the
numerator.
1
4x
56.
, x2
, x 0, 4
2
x4
x1
58.
xx 1, x 1, 0
, x0
x1
2x 1
1
x7 2
60. 2 2
61.
, x > 0
2x
x2
t t 1
77.
79.
80.
82.
83.
x8 5
1
2xx 5
63. 2
64.
x3
x 15
x 54
3
2
3
2
2x 2x 5
4x 2x 1 2x
66.
x 112
2x 112
3x 1
x2 2
68. 3
, x0
3
x 1 x 212
1
2x h
70. 2
, h0
, h0
xx h
x x h2
1
, h0
x 4x h 4
1
1
73.
, h0
x h 1x 1
x 2 x
1
1
75.
, h0
z 3 z
x h 1 x 1
1
, h0
x h 2 x 2
x
8x 2
78.
, x0
22x 1
x 42
1
x
60 15
(a)
minute
(b)
minute(s)
(c)
minutes
16
4
16
16
8t
288MN P
81. (a) 9.09%
(b)
; 9.09%
15
NMN 12P
288MN P
(a) 4.57%
(b)
; 4.57%
NMN 12P
(a)
t
0
2
4
6
8
10
T
75
55.9
48.3
45
43.3
42.3
t
12
14
16
18
20
22
T
41.7
41.3
41.1
40.9
40.7
40.6
(b) The model is approaching a T-value of 40.
84. (a)
Year, t
Banking
Paying bills
(millions)
(millions)
2002
21.9
13.9
2003
27.0
17.8
2004
31.4
21.5
2005
35.6
25.0
2006
40.2
28.1
2007
45.6
31.0
(b) The estimates are fairly close to the actual numbers of
households.
(c)
0.2151t 3 2.408t 2 7.75t 5.5
Rate 0.00146t 4 0.0403t 3 0.491t 2 23.81t 0.3