algebra 2 honors: final exam review
TRANSCRIPT
Name: ________________________ Class: ___________________ Date: __________
Algebra 2 Honors: Final Exam Review
Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt all problems!!!! You will be able to use a calculator on the exam.
1
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1. Find the product 5x − 3( )(x3 − 5x + 2).
a. 5x4 − 3x3 − 25x2 + 25x − 6 b. 5x3 + 22x2 − 5x − 6 c. 5x3 − 28x2 + 25x − 6 d. 5x4 − 3x3 + 25x2 − 5x − 6
Simplify the given expression.
2. 5(a 2 + 5a + 6)
3(a2 − 36)÷
41(a + 3)6(a + 6)
a. 10(a + 3)(a − 2)
41(a − 6) b.
10(a + 2)41(a − 6)
c. 10(a + 2)41(a + 6)
d. 10(a + 3)(a + 2)41(a + 6)(a − 6)
3. 12x2y
⋅3y2
24x3
a. 3y4x
b. 3y2
4x2 c. 3y
4x2 d. 4y
3x2
4. Which gives the solution(s) of the equation x − 53 = −6?a. –211, 221 b. 41 c. 221 d. –211
Solve the given equation.
5. 9x − 9 + 5 = 10
a. 1049
b. 349
c. 1099
d. 149
6. 2 9n − 11 =1
16
a. n = 79
b. n = 53
c. n = 89
d. n = 7
7. 625x − 4 = 25x + 5
a. 14 b. 13 c. –13 d. –3
8. Solve the equation. −x2 + 4 = 2x2 − 5
Factor the polynomial completely.
9. 30x3 − 50x2 + 27x − 45a. 10x2(3x − 5) − 9(3x − 5) b. (10x2 + 9)(3x − 5) c. 10x2(3x − 5) − 27x + 45 d. (30x3 − 50x2) + (27x − 45)
10. 8a4b 2 − 12a3b2
a. 4(2a4b2 − 3a3b 2) b. a3b2(8a − 12) c. 4a3b 2(2a − 3) d. 4a2b2(2a2 − 3)
2
Graph:
11. Which is the graph of f(x) = x + 13 − 5?
a. c.
b. d.
12. Evaluate the expression log3
127
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃ .
a. −13
b. −3 c. 13
d. 3
13. Identify the vertex and the axis of symmetry of the graph of the function y = 2(x + 2)2 − 4.a. vertex: (–2, 4);
axis of symmetry: x = −2b. vertex: (2, –4);
axis of symmetry: x = 2c. vertex: (–2, –4);
axis of symmetry: x = −2d. vertex: (2, 4);
axis of symmetry: x = 2
14. The price per person of renting a bus varies inversely with the number of people renting the bus. It costs $15 per person if 44 people rent the bus. About how much will it cost per person if 71 people rent the bus?a. $9.30 b. $24.20 c. $208.27 d. $6.21
Solve for x.
15. 3x2 = 147
a. ±21 b. ± 144 c. ± 441 d. ±7
3
16. Graph the rational function f x( ) = 1x− 1
. Then find its domain and range.
a.
Domain: all real numbers except 1Range: all real numbers except 1
c.
Domain: all real numbers except –1Range: all real numbers except 1
b.
Domain: all real numbers except –1Range: all real numbers except 0
d.
Domain: all real numbers except 1Range: all real numbers except 0
Simplify the given expression. Assume that no variable equals 0.
17. 19x−6y11ÊËÁÁÁ
ˆ¯˜̃̃ −6xy5ÊËÁÁÁ
ˆ¯˜̃̃
a. −114x−5y16 b. 13y16
x5 c. −114y16
x5 d. −114x−7y−24
18. 32x18y10
16x9y20
Ê
Ë
ÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃˜̃̃
2
a. 2x9y20 b. 4x18
y20 c. 4x9
y10 d. 4x18y−20
Simplify the expression.
19. 24564
a. 7 5
8 b.
498
c. 5
8 d.
7 78
20. 11+ i( ) + 3− 15i( )
a. 14− 14i b. − 4+ 4i c. 12− 12i d. 14+ 16i
21. −4+ 4i( )(−3− 3i)
a. 16+ 12i b. 12+ 0i − 12i2 c. 24+ 0i d. 12+ 0i + 12
4
22. Divide x3 + x2 − x + 2 by x + 4.a. x2 − 3x + 11, R –42 b. x2 − 3x + 11 c. x2 + 5x − 13 d. x2 + 5x − 13, R 46
23. Divide 5x3
3x2y÷ 25
3y9 . Assume that all expressions are defined.
a. xy8
5 b.
125x
9y10 c. x
5y8 d. 5
xy8
24. Suppose that x and y vary inversely, and x = 2 when y = 8. Write the function that models the inverse variation.
a. y =16
x b. y =
10
x c. y = 4x d. y =
6
x
25. Solve 8x + 8 = 32x .a. x = –12 b. x = 22 c. x = 12 d. x = –22
26. Identify the maximum or minimum value and the domain and range of the graph of the function
y = 2(x + 2)2 − 3.a. minimum value: 3
domain: all real numbers ≥ 3range: all real numbers
b. maximum value: –3domain: all real numbers ≤ −3range: all real numbers
c. maximum value: 3domain: all real numbersrange: all real numbers ≤ 3
d. minimum value: –3domain: all real numbersrange: all real numbers ≥ −3
27. Which is an equation for the inverse of the relation y = 4x + 2?
a. y = 2x + 4 b. y = 4x − 24
c. y = x + 24
d. y = x − 24
What are the solutions?
28. 9x2 + 16 = 0
a. −43
i, 43
i b. −169
i, 169
i c. −34
i, 34
i d. −43
, 43
29. Use inverse operations to write the inverse of f(x) = x
4 – 5.
a. f −1(x) = 4x + 5 b. f −1(x) = –5x − 4 c. f −1(x) = 4(x + 5) d. f −1(x) = x
4 + 5
30. Write the equation of the parabola y = x2 − 4x − 15 in standard form.
a. y + 4 = x − 2( )2 b. y + 19= x − 2( )
2 c. y − 2 = x − 2( )2 d. y + 15= x − 2( )
2
5
31. Consider the function f(x) = −4x2 − 8x + 10. Determine whether the graph opens up or down. Find the axis of symmetry, the vertex and the y-intercept. Graph the function.a. The parabola opens downward.
The axis of symmetry is the line x = −1.The vertex is the point (−1,14).The y-intercept is 10.
c. The parabola opens upward.The axis of symmetry is the line x = −1.The vertex is the point (−1,−6).The y-intercept −5.
b. The parabola opens upward.The axis of symmetry is the line x = −1.The vertex is the point (−1,14).The y-intercept 10.
d. The parabola opens downward.The axis of symmetry is the line x = −1.The vertex is the point (−1,7).The y-intercept is 5.
Simplify.
32. 3+ 5
4− 5
a. 17+ 7 5
11 b.
17+ 7 5
2 c.
17− 7 511
d. 17− 7 5
2
33. 192 − 245 + 27 + 80
a. 11 3 − 3 5 b. 11 3 + 3 5 c. 3 3 − 11 5 d. 3 3 + 11 5
6
Divide.
34. 2x4 − 4x3 − 12x − 15ÊËÁÁÁ
ˆ¯˜̃̃ ÷ x − 3( )
a. 2x2 + 2x + 6 +3
x − 3 b. 2x2 + 2x + 6 + 3 c. 2x3 + 2x2 + 6x + 6 + 3 d. 2x3 + 2x2 + 6x + 6 +
3x − 3
35. Tell whether the function y = 2 5( )x shows growth or decay. Then graph the function.
a. This is an exponential growth function.c. This is an exponential decay function.
b. This is an exponential growth function. d. This is an exponential growth function.
36. 16x2 − 25a. (4x − 5)2 b. (4x + 5)(−4x − 5) c. (4x + 5)(4x − 5) d. (−4x + 5)(4x − 5)
Solve the equation. Check for extraneous solutions.
37. 3
k 2 − 1= 3
k + 1a. 3 b. 0 c. 2 d. 1
7
Find the sum or difference.
38. 6q5 + 8q 2 + 3ÊËÁÁÁ
ˆ¯˜̃̃ + 8q5 − 3q − 7Ê
ËÁÁÁ
ˆ¯˜̃̃
a. −2q5 + 8q2 − 3q + 10 b. 14q 5 + 8q 2 − 3q − 4 c. −2q5 + 8q2 + 3q + 10 d. 14q5 + 5q2 − 4
Solve the system by graphing.
39. −2x = 3y − 2
x − y = −4
Ï
ÌÓ
ÔÔÔÔÔÔÔÔÔÔ
a.
(–2, –2)
c.
(2, –2)b.
(–2, –2)
d.
(–2, 2)
What is the quotient in simplified form? State any restrictions on the variable.
40. x2 − 16
x2 + 5x + 6÷
x2 + 5x + 4
x2 − 2x − 8
a. (x − 4)2
(x + 3)(x + 1) b.
(x + 4)2(x + 1)
(x + 2)2(x + 3) c.
(x − 4)2
(x + 3)(x + 1) d.
1
(x + 3)(x + 1)
8
Solve the equation.
41. 5n − 2 n − 2( ) = −11
42. 14
3y + 2ÊËÁÁ ˆ
¯˜̃ = 7
43. 3x + 17− 5x = 12− 6x + 3( )
44. –3x + 25 + x + 21 = 2a. 22 b. –3 c. –22 d. 3
What is the graph of the rational function?
45. y =−3x + 5
−5x + 2a. b.
c. d.
Solve the equation.
46. x3 + 4x2 − 25x − 100= 0a. −4, 25 b. −5, 4, 5 c. 4, 5 d. −4, − 5, 5
47. x3 − x2 − 6x = 0
9
Solve the equation. Check for extraneous solutions.
48. log5 3x + 9( ) = 2
a. 233
b. 163
c. 13
d. 343
49. Add. Write your answer in standard form.
(4d5 − d3) + (d 5 + 6d3 − 4)
a. 5d5 + 5d3 − 4 b. 5d5 + 5d3 c. 5d10 + 5d6 − 4 d. 4d5 + 6d3 − 4
50. Which is the graph of the function f(x) = x3 − 2x?
a. b.
c. d.
51. Add x + 6x − 7
+ −12x − 59
x2 − 3x − 28.
a. −11x − 53
x2 − 2x − 35 b.
x2 + 10x + 24(x + 4)(x − 7)
c. x + 6
(x − 7)(x + 4) d.
x + 5x + 4
52. Express log264− log24 as a single logarithm. Simplify, if possible.
a. log24 b. 8 c. 4 d. log260
10
What is the expression in factored form?
53. 16x2 + 8x
a. −4x(4x + 2) b. 4x(4x − 2) c. 4x(4x + 2) d. 4(4x + 2)
54. Identify the axis of symmetry for the graph of f(x) = 4x2 − 8x + 3.a. x = −1 b. y = −1 c. y = 1 d. x = 1
Solve the linear system.
55. −4x − 3y = −27−4x + 4y = 8a. (–5, –5) b. (–1, –5) c. (3, 5) d. no solution
56. List all of the possible rational zeros of the following function.
f x( ) = 2x6 − 10x5 − 23x4 + 80x3 + 28x2 − 20x + 9
a. ±1, ±3, ±9, ±12
, ±32
, ±92
b. 1, 3, 9, ±12
, ±32
, ±92
c. 1, 3, 9, 12
, 32
, 92
d. –1, –3, –9, −12
, −32
, −92
Determine the solution of the system of inequalities.
57. y ≤ −x − 1−2x + y ≥ −2a. b.
c. d.
11
Factor completely.
58. 60z6 − 118z5 + 56z4
a. 2z4(5z − 7)(6z − 4) b. z4(6z − 7)(5z − 4) c. z4(5z + 7)(6z − 4) d. 2z4(6z − 7)(5z − 4)
59. Find the y-intercept of the equation. y = − 3 ⋅ 7x
a. 4 b. –21 c. –3 d. 7
Solve the following system of equations by graphing.
60. 2y + 8x = 58y − 5x = 11a. (2, 21) b. (21, 2) c. (4, 20) d. (1, 21)
61. Find the zeros of f(x) = x2 + 7x + 9 by using the Quadratic Formula.
a. x =−7± 13
2 b. x = −7± 13 c. x =
3± 72
d. x = 3± 7
62. Simplify the expression 256z164 . Assume that all variables are positive.
a. 2564 z4 b. 4z4 c. 2564 z11 d. 4z11
63. Which represents the graph of y = x + 4 ? State the domain and range of the function.
a.
Domain:x ≥ −4; Range:y ≥ 0
c.
Domain:x ≥ 4; Range:y ≥ 0
b.
Domain:x ≥ 0; Range:y ≥ −4
d.
Domain:x ≥ 0; Range:y ≥ 4
12
Solve the given equation. If necessary, round to four decimal places.
64. 92x = 21a. 11.0035 b. 3.0445 c. 2.1972 d. 0.6928
Sketch the asymptotes and graph the function.
65. y =5
x − 3+ 2
a. b.
c. d.
66. Solve x4 − 3x3 − x2 − 27x − 90= 0 by finding all roots.a. The solutions are 5 and −2. b. The solutions are 5, −2, 3i, and −3i. c. The solutions are −3, −1, −27, and −90. d. The solutions are −5, 2, 3i, and −3i.
Divide the expressions. Simplify the result.
67. x2 + 9x + 18
x2 − 9 ÷
x + 6x − 6
a. 9x + 6
3 b.
x − 6x − 3
c. x − 9x − 3
d. x + 3x − 6
68. Simplify log7x3 − log7x .
a. log7(x3 − x) b. 2log7x c. log72x d. 2(x3 − x)
13
Solve the equation by factoring.
69. 2x2 + 3x − 14= 0
a. {–4, −72
} b. {−72
, 2} c. {–4, 7} d. {2, 7}
Solve the system of inequalities by graphing.
70. y ≤ −3x − 1
y > 3x − 2
Ï
ÌÓ
ÔÔÔÔÔÔÔÔÔÔ
a. b.
c. d.
71. Multiply 8x4y2
3z3 ⋅9xy2z6
4y4 . Assume that all expressions are defined.
a. 6x4yz2 b. 6x5y8z9 c. 6x5z3 d. 32 x3y2z
72. Solve the equation 6x
x − 3= 4x + 6
x − 3.
a. There is no solution. b. x = − 3
2 c. x = 3 d. x = −3
73. Which is equivalent to 81−1/4?
a. 9 b. 3 c. 19
d. 13
74. Which gives the solution(s) of x + 72 = x?a. –8 b. 9 c. no solution d. 9, –8
14
75. Find the zeros of the function h x( ) = x2 + 23x + 60 by factoring.a. x = −20 or x = −3 b. x = 4 or x = 15 c. x = −4 or x = −15 d. x = 20 or x = 3
What are the zeros of the function? Graph the function.
76. y = (x + 3)(x − 3)(x − 4)a. 3, –3, –4 c. 3, –3, 4
b. –3, 3, 4 d. –3, 3, –4
77. Solve the equation 2x2 + 18= 0.a. x = ±3i b. x = ±3 c. x = 3± i d. x = ±3+ i
78. Use substitution to solve the system 3x + y = −3
y = x + 5
Ï
ÌÓ
ÔÔÔÔÔÔÔÔÔÔ
.
a. (− 8
3 , –3) b. (−2, 3) c. (− 4
3 , 1) d. (3, −2)
79. Find the value of f(–9) and g(–2) if f x( ) = −5x − 2 and g x( ) = 3x2 − 21x .a. f(–9) = –7
g(–2) = 19c. f(–9) = 43
g(–2) = 54b. f(–9) = 47
g(–2) = –83d. f(–9) = 10
g(–2) = –27
Determine the value or values of the variable where the expression is not defined.
80. x − 7
x2 − 6x − 16
15
What is the solution of each equation?
81. 108x2 = 147
a. −4936
, 4936
b. −76
, 76
c. −67
, 67
d. −3649
, 3649
82. Simplify 10− x2 − 3x
x2 + 2x − 8. Identify any x-values for which the expression is undefined.
a.−x − 5x + 4
The expression is undefined at x = −4.
c.x + 5x + 4The expression is undefined at x = 2 and x = −4.
b.−x − 5x + 4
The expression is undefined at x = 2 and x = −4.
d.x + 5x + 4The expression is undefined at x = −4.
83. Determine whether the binomial (x − 4) is a factor of the polynomial P x( ) = 5x3 − 20x2 − 5x + 20.
a. (x − 4) is not a factor of the polynomial P x( ) = 5x3 − 20x2 − 5x + 20. b. (x − 4) is a factor of the
polynomial P x( ) = 5x3 − 20x2 − 5x + 20. c. Cannot determine.
84. Express as a single logarithm: loga 13+ loga 60
a. loga 780 b. loga 13+60( ) c. loga
1360
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃ d. loga 1360
85. Graph the system of inequalities y < −3x + 2
y ≥ 4x − 1
Ï
ÌÓ
ÔÔÔÔÔÔÔÔÔÔ
.
a. b.
c. d.
16
86. Solve using factoring: 3x2 + 5x − 12= 0
a. −43
, 3 b. 4, − 3 c. 8, − 18 d. 43
, − 3
87. Solve x2 + x − 30
x − 5= 11. Check your answer.
a. x = 5 b. x = 16 c. x = −6 d. There is no solution because the original equation is undefined at x = 5.
Perform the indicated operation(s) and simplify.
88. 4
x + 8 +
1x − 8
a. 5
x + 8 b.
5
x2 − 64 c.
5x− 245
d. 5x− 24
x2 − 64
89. Subtract −6x2 + x − 3
x2 + 9− −2x − 4
x2 + 9. Identify any x-values for which the expression is undefined.
a. −6x2 − x − 7
x2 + 9; The expression is always defined. b.
−6x2 − x − 7
x2 + 9; The expression is undefined at x = ±3.
c. −6x2 + 3x + 1
x2 + 9; The expression is undefined at x = ±3. d.
−6x2 + 3x + 1
x2 + 9;
The expression is always defined.
90. Which is the graph of f x( ) = x − 2x − 1
?
a. c.
b. d.
91. Divide by using synthetic division.
(x2 − 9x + 10)÷ x − 2( )
a. x − 9+ 6
x − 2 b. x − 11+ 32
x − 2 c. x − 7+ −4
x − 2 d. 2x − 18+ 10
x − 2
17
92. If y varies inversely as x and y = 132 when x = −18, find y when x = 50. Round your answer to the nearest hundredth, if necessary.a. 47.52 b. 366.67 c. –47.52 d. –366.67
Graph:
93. f(x) = 214
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃
x
a. c.
b. d.
94. Simplify the expression (27)
1
3 ⋅ (27)
2
3.
a. 9 b. 3 c. 27 d. 729
What are the solutions of the following systems?
95. −x + 2y = 10
−3x + 6y = 11
Ï
ÌÓ
ÔÔÔÔÔÔÔÔÔÔ
a. infinitely many solutions b. (–5, 2) c. (5, –2) d. no solutions
Simplify:
96. −3x + 3x2
−24x + 24
a. −x8
b. x − x2
8x − 8 c.
x2
16 d.
1− x16
Find the inverse of the given function.
97. f x( ) = 7x − 316
a. f −1 x( ) = 16x − 37
b. f −1 x( ) = 16x + 37
c. f −1 x( ) = 7x + 163
d. f −1 x( ) = 7x − 163
18
98. Use a graph to solve the system −5x + 4y = 6
3x − y = 2
Ï
ÌÓ
ÔÔÔÔÔÔÔÔÔÔ
. Check your answer.
a.
The solution to the system is (2, 4).
c.
The solution to the system is (–2, 4).b.
The solution to the system is (–2, –4).
d.
The solution to the system is (2, –4).
Simplify the rational expression, if possible.
99. n2 + 2n − 24
n2 − 11n + 28
a. n + 6n + 7
b. n + 6n − 7
c. n + 6n − 4
d. n − 4n − 7
100. Write a quadratic function in standard form with zeros 6 and –8.
a. f(x) = x2 + 2x − 48 b. f(x) = x2 − 2x − 48 c. f(x) = x2 − 4x + 4 d. 0 = x2 + 2x − 48
101. Let f(x) = x2 − 5 and g(x) = 3x2 . Find g(f(x)).
a. 3x4 − 30x2 + 75 b. 3x4 − 15 c. 3x4 − 5 d. 9x4 − 5
102. Simplify 84 / 3.
a. 12
b. 8 c. 323
d. 16
Solve.
103. x2 − 6x = 0a. 0, 6 b. 0, –6 c. –6, 6 d. 1, 6
19
104. Graph the function. Label the vertex, axis of symmetry, and x-intercepts.
y = 2 x + 2( ) x + 4( )
a.
vertex: (−3, −2)axis of symm: x = −3x-intercepts: –4, –2
c.
vertex: (3, −2)axis of symm: x = 3x-intercepts: 2, 4
b.
vertex: (−3, 2)axis of symm: x = −3x-intercepts: –4, –2
d.
vertex: (3, 2)axis of symm: x = 3x-intercepts: 2, 4
Find the roots of the polynomial equation.
105. x3 − 2x2 + 10x + 136 = 0a. –3 ± 5i, –4 b. 3 ± 5i, –4 c. –3 ± i, 4 d. 3 ± i, 4
106. Solve log3x = 6.
a. 18 b. 216 c. 6 d. 729
Simplify the sum.
107. w2 + 2w − 24
w2 + w − 30+
8
w − 5
a. w − 4
w − 5 b.
w2 + 2w − 16
w2 + w − 30 c. w + 4 d.
w + 4
w − 5
108. Factor out the greatest common monomial factor from 21z3 + 28z.
20
109. Nadav invests $6,000 in an account that earns 5% interest compounded continuously. What is the total amount of her investment after 8 years? Round your answer to the nearest cent.a. $8950.95 b. $327,588.90 c. $9850.95 d. $14,950.95
110. Write the logarithmic equation log416= 2 in exponential from.
a. 416 = 2 b. 42 = 16 c. 2−4 = 16 d. 24 = 16
111. Solve the equation x2 − 10x + 25= 54.
a. x = 5−3 6 b. x = 5± 6 3 c. x = 5±3 6 d. x = 5+3 6
Solve.
112. 3 x + 7( )2 + 17 = 59
a. −7 ± 13 b. 42 ± 14 c. 42 ± 13 d. −7 ± 14
113. 8x2 + 23= 823
a. no real-number solution b. ± 800 c. ± 64 d. ±10
114. 3x2 − 9 = 3
Solve by using tables. Give each answer to at most two decimal places.
115. 2x2 + 5x − 3 = 0a. 0.5, –3 b. 1, –6 c. 3, –0.5 d. 1.75, –1.75
116. Solve the polynomial equation 3x5 + 6x4 − 72x3 = 0 by factoring.a. The roots are –6 and 4. b. The roots are 0, –6, and 4. c. The roots are 0, 6, and –4. d. The roots are –18 and 12.
Simplify the given expression.
117. 3
4x2 − 25 +
22x + 5
a. 4x + 7
(2x + 5)(2x − 5) b.
4x − 10(2x − 5)(2x + 5)
c. 4x − 7
(2x + 5)(2x − 5) d.
5
(4x2 + 2x − 20)
Solve the equation. Check the solution.
118. g + 4
g − 2=
g − 5
g − 8
a. −223
b. 22 c. −22 d. 14
Solve.
119. x2 − 10x + 29= 0a. −5+ 4i, −5− 4i b. −5+ 2i, −5− 2i c. 5+ 4i, 5− 4i d. 5+ 2i, 5− 2i
21
Find the quotient.
120. x + 4x − 4
÷ x2 − 164− x
a. x + 4x − 4
b. 1
4− x c.
1x − 4
d. 1
2− x
Evaluate each expression.
121. 4log48.2
a. 8.24 b. 8.2 c. 4 d. 48.2
122. Simplify the expression log464.
a. 16 b. 64 c. 4 d. 3
123. If $2500 is invested at a rate of 11% compounded continuously, find the balance in the account after 4 years.
Use the formula A =Pe rt .a. $3795.18 b. $3881.77 c. $4333.13 d. $18472.64
Simplify:
124. 12 + 48
a. 6 3 b. 60 c. 24 3 d. 3 6
Multiply the expressions. Simplify the result.
125. d2
ef·
5e5f4d
a. 5de4
4 b.
54
c. 4d 3
5e6f 2 d. 5d2e5
4def
126. Is (x − 2) a factor of P(x) = x3 + 2x2 − 6x − 4? If it is, write P(x) as a product of two factors.a. yes:
P(x) = (x + 2)(x2 + 4x + 2)c. yes:
P(x) = (x − 2)(x2 − 4x + 2)b. yes:
P(x) = (x − 2)(x2 + 4x + 2)d. (x − 2) is not a factor of P(x)
Write the expression as a complex number in standard form.
127. (−3− 8i) + (−5− 7i)a. 2+ 15i b. 2− 15i c. −8+ 15i d. −8− 15i
128. 2+ 3i( ) 1− 4i( )
ID: A
1
________________________________________________________________Answer Section
1. A
5x − 3( )(x3 − 5x + 2)
= 5x(x3 − 5x + 2) − 3(x3 − 5x + 2) Distribute 5x and −3.
= 5x(x3) + 5x(−5x) + 5x 2( ) − 3(x3) − 3 −5x( ) − 3 2( ) Distribute 5x and −3 again.
= 5x4 − 25x2 + 10x − 3x3 + 15x − 6 Multiply.
= 5x4 − 3x3 − 25x2 + 25x − 6 Combine like terms.
2. BTo divide two rational expressions, multiply the first expression by the reciprocal of the divisor.
3. CFirst, multiply the values and then divide the numerator and the denominator by the common factors.
4. D 5. B
Isolate the radical in the original equation, and raise each side of the equation to the power equal to the index of the radical to eliminate the radical. Check the solution obtained by substituting the value of x in the original equation.
6. AEliminate the bases and use the Property of Equality for Exponential Functions to solve the equation.
7. B
8. x = ± 3 9. B
Group the monomials to find the GCF (greatest common factor), factor the GCF of each binomial, and then use the Distributive Property to obtain the factors.
10. CFind the GCF (greatest common factor) of the monomials in the given polynomial, and use it in grouping the polynomial.
11. B 12. B 13. C 14. A 15. D 16. D 17. C
Multiply the constants and then multiply the powers using the Power of a Product Property.
18. BSimplify each base using the properties of powers. Then, write all the fractions in the simplest terms and ensure there are no negative exponents.
19. A
ab
=a
b
20. ACombine the real and imaginary parts of the complex numbers to add them.
ID: A
2
21. C
Use the FOIL method to multiply the complex numbers and use the formula i2 = −1. Combine the real parts and then the imaginary parts of the two numbers.
22. A 23. A
5x3
3x2y÷ 25
3y9 Rewrite as multiplication by the reciprocal.
= 5x3
3x2y⋅3y9
25Simplify by canceling common factors.
= xy8
5
24. A 25. C
23ÊËÁÁÁ
ˆ¯˜̃̃
x + 8
= 25ÊËÁÁÁ
ˆ¯˜̃̃
x
Rewrite each side as powers of the same base.
23(x + 8) = 25x To raise a power to a power, multiply the exponents.
3(x + 8) = 5x The bases are the same, so the exponents must be equal.x = 12
The solution is x = 12.
26. D 27. D 28. A 29. C
f(x) = x
4 + (–5) The variable, x, is divided by 4, then –5 is added.
f −1(x) = 4(x – (–5))Undo the addition by subtracting –5. Undo the division by multiplying by 4.
f −1(x) = 4(x + 5)
30. B 31. A
Because a is −2, the graph opens downward.
The axis of symmetry is given by x =−(−8)
2(−4) = 8
−8 = −1.
x = −1 is the axis of symmetry.
The vertex lies on the axis of symmetry, so x = −1.The y-value is the value of the function at this x-value.
f(−1) = −4(−1)2 − 8(−1) + 10= −4+ 8+ 10= 14The vertex is (−1,14).Because the last term is 10, the y-intercept is 10.
32. A
Multiply the numerator and denominator by the conjugate of the denominator and simplify.
33. AFind the principal square root of each term of the radicand and simplify the expression.
ID: A
3
34. D 35. B
Step 1 Find the value of the base: 5.The base is greater than 1. So, this is an exponential growth function.
Step 2 Choose several values of x and generate ordered pairs. Then, graph the ordered pairs and connect with a smooth curve.
36. C 37. C 38. B 39. D 40. C 41. –5
42. 263
43. –2 44. A 45. A 46. D 47. −2, 0, 3 48. B 49. A
(4d5 − d3) + (d 5 + 6d3 − 4)
= (4d5 + 6d3) + (−d3 + d5) + −4( )Identify like terms. Rearrange terms to get like terms together.
= 5d5 + 5d3 − 4 Combine like terms.
50. B 51. D
x + 6x − 7
+ −12x − 59(x + 4)(x − 7) Factor the denominators. The LCD is (x + 4)(x − 7).
= x + 4x + 4
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃
x + 6x − 7
+ −12x − 59(x + 4)(x − 7)
Multiply by x + 4x + 4
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃ .
= x2 + 10x + 24(x + 4)(x − 7)
+ −12x − 59(x + 4)(x − 7)
= x2 − 2x − 35(x + 4)(x − 7)
Add the numerators.
= (x + 5)(x − 7)(x + 4)(x − 7)
Factor the numerator.
= x + 5x + 4
Divide the common factor.
52. CTo subtract the logarithms divide the numbers.
log264− log24 = log2( 64
4 ) = log216 = 4
53. C
ID: A
4
54. D
If a function has one zero, use the x-coordinate of the vertex to find the axis of symmetry.If a function has two zeros, use the average of the two zeros to find the axis of symmetry.
55. C 56. A
Use the Rational Zero Theorem. 57. C 58. D 59. C 60. A
Graph the equations and find their point of intersection.
61. A
x2 + 7x + 9 = 0 Set f(x) = 0.
x =−b ± b 2 − 4ac
2aWrite the Quadratic Formula.
x =−7± (7)2 − 4(1)(9)
2(1)Substitute 1 for a, 7 for b, and 9 for c.
x =−7± 49− 36
2Simplify.
x =−7± 13
2Write in simplest form.
62. B
256z164
= 256 ·z4 · z4 · z4 · z44 Factor into perfect powers of four.
= 4· z · z · z · z Use the Product Property of Roots.
= 4z4 Simplify.
63. A 64. D
Use the Property of Inequality for Logarithmic Functions and the Power Property of Logarithms to solve the equation.
65. C
ID: A
5
66. BThe polynomial is of degree 4, so there are four roots for the equation.Step 1: Identify the possible rational roots by using the Rational Root Theorem.±1,±2,±3,±5,±6,±9,±10,±15,±18± 30,±45,±90
±1p = −90 and q = 1
Step 2: Graph x4 − 3x3 − x2 − 27x − 90= 0 to find the locations of the real roots.
The real roots are at or near 5 and −2.
Step 3: Test the possible real roots.Test the possible root of 5:
5| 1 −3 −1 −27 −90
5 10 45 90
1 2 9 18 0
Test the possible root of −2:−2| 1 −3 −1 −27 −90
−2 10 −18 90
1 −5 9 −45 0
The polynomial factors into x − 5( ) x + 2( )(x2 + 9) = 0.
Step 4: Solve x2 + 9 = 0 to find the remaining roots.
x2 + 9 = 0
x2 = −9
x = ±3iThe fully factored equation is x − 5( ) x + 2( ) x + 3i( ) x − 3i( ) = 0.The solutions are 5, −2, −3i, and 3i.
67. B 68. B
log7x3 − log7x= 3log7x − log7x Use the Power Property of Logarithms. = 2log7x Simplify.
69. BFor any real numbers a and b, if ab = 0, then either a = 0, b = 0, or both a and b are equal to zero.
70. D
ID: A
6
71. C
Arrange the expressions so like terms are together: 8 ⋅ 9(x4 ⋅ x)(y2 ⋅ y2)z6
3 ⋅ 4 ⋅ z3y4 .
Multiply the numerators and denominators, remembering to add exponents when multiplying: 72x5y4z6
12z3y4 .
Divide, remembering to subtract exponents: 6x5y0z3.
Since y0 = 1, this expression simplifies to 6x5z3.
72. A6x
x − 3(x − 3) = 4x + 6
x − 3(x − 3) Multiply each term by the LCD, (x – 3).
6x = 4x + 6 Simplify. Note that x g 3.2x = 6 Solve for x.x = 3
The solution x = 3 is extraneous because it makes the denominators of the original equation equal to 0. Therefore the equation has no solution.
73. D 74. B 75. A
h x( ) = x2 + 23x + 60
x2 + 23x + 60 = 0 Set the function equal to 0.(x + 20)(x + 3) = 0 Factor: Find factors of 60 that add to 23.x + 20 = 0or x + 3 = 0 Apply the Zero-Product Property.x = −20or x = −3 Solve each equation.
76. B 77. A
2x2 + 18= 02x2 = −18 Add −18 to both sides.
x2 = −9 Divide both sides by 2.
x = ± −9 Take square roots.
x = ±3i Express in terms of i.
78. BStep 1 y = x + 5 The second equation is solved for y.Step 2 3x + y = −3
3x + x + 5( ) = −3 Substitute x + 5 for y in the first equation.Step 3 4x + 5 = −3 Simplify and solve for x.
4x = −84x4
= −84
Divide both sides by 4.
x = −2Step 4 y = x + 5 Write one of the original equations.
y = −2 +5 Substitute −2 for x.y = 3 Find the value of y.
(−2, 3) Write the solution as an ordered pair.
ID: A
7
79. CSubstitute x = –9 in the equation f(x) and x = –2 in the equation g(x).
80. x ≠ 8, x ≠ −2 81. B 82. B
−1(x2 + 3x − 10)
x2 + 2x − 8Factor −1 from the numerator and reorder the terms.
= −1(x + 5)(x − 2)(x + 4)(x − 2) Factor the numerator and denominator.
= −x − 5x + 4
Divide the common factors and simplify.
The expression is undefined at those x-values, 2 and −4, that make the original denominator 0.
83. BFind P 4( ) by synthetic substitution.
4 5 −20 −5 20
20 0 −20
5 0 −5 0
Since P 4( ) = 0, x − 4 is a factor of the polynomial P x( ) = 5x3 − 20x2 − 5x + 20.
84. A 85. A
Graph y < −3x + 2 and y ≥ 4x − 1 on the same coordinate plane. The solutions of the system are the overlapping shaded regions, including the solid boundary line.
86. D 87. D
x2 + x − 30x − 5
= 11 Note that x ≠ 5.
x − 5( ) x + 6( )
x − 5= 11 Factor.
x + 6 = 11 The factor x − 5( ) cancels.x = 5Because the left side of the original equation is undefined when x = 5, there is no solution.
88. D 89. D
−6x2 + x − 3
x2 + 9− −2x − 4
x2 + 9
= −6x2 + x − 3+ 2x + 4
x2 + 9Subtract the numerators. Distribute the negative sign.
= −6x2 + 3x + 1
x2 + 9Combine like terms.
There is no real value of x for which x2 + 9 = 0; the expression is always defined.
90. D
ID: A
8
91. CFor x − 2( ), a = 2.
2 1 –9 10 Write the coefficients of the expression. Bring down the first coefficient. Multiply and add each column.
2 –141 –7 –4
Write the remainder as a fraction to get x − 7+ −4
x − 2 .
92. CUse inverse proportion to relate values in the following equation.x1
y2
=x2
y1
93. D 94. C
(27)
1
3 ⋅ (27)
2
3
= (27)
1 + 2
3 Product of Powers
= (27)1 Simplify.
= 27
95. D 96. A 97. B
The inverse function can be found by exchanging the domain and range of the function.
98. ASolve each equation for y.
y = 5
4 x + 3
2
y = 3x − 2
Ï
Ì
Ó
ÔÔÔÔÔÔÔÔÔÔÔÔ
Then graph each equation.The lines appear to intersect at the point (2, 4). Check by substituting the x- and y-values into each equation.
99. B 100. A
x = 6or x = −8 Write the zeros as solutions for two equations.x − 6 = 0or x + 8 = 0 Rewrite each equation so that it is equal to 0.
0 = (x − 6)(x + 8) Apply the converse of the Zero-Product Property to write a product that is equal to 0.
0 = x2 + 2x − 48 Multiply the binomials.
f(x) = x2 + 2x − 48 Replace 0 with f(x)
101. A 102. D 103. A 104. A 105. B 106. D
Use the definition of logarithms with base b.
107. D
ID: A
9
108. 7z 3z2 + 4ÊËÁÁÁ
ˆ¯˜̃̃
109. A
A = Pe rt Substitute 6,000 for P, 0.05 for r, and 8 for t.
A = 6000e0.05 8( ) Use the [e^x] key on a calculator.A ≈ 8950.95
The total amount after 8 years is $8950.95. 110. B
Logarithmic form: log416= 2
Exponential form: 42 = 16The base of the logarithm becomes the base of the power, and the logarithm is the exponent.
111. C
x2 − 10x + 25= 54x − 5( )
2 = 54 Factor the perfect square trinomial.
x − 5 = ± 54 Take the square root of both sides.
x = 5± 54 Add 5 to each side.
x = 5±3 6 Simplify.
112. D 113. D 114. ±2 115. A 116. B
3x5 + 6x4 − 72x3 = 0
3x3 x2 + 2x − 24ÊËÁÁÁ
ˆ¯˜̃̃ = 0
Factor out the GCF, 3x3.
3x3 x + 6( ) x − 4( ) = 0 Factor the quadratic.
3x3 = 0, x + 6 = 0, x − 4 = 0 Set each factor equal to 0.x = 0, x = −6, x = 4 Solve for x.
117. CFind equivalent fractions that have a common denominator. Then, simplify each numerator and denominator and add the numerators.
118. D 119. D 120. B 121. B 122. D
Factor 64. Then write it in the form of 43, and apply the Inverse Properties of Logarithms and Exponents. 123. B 124. A 125. A 126. B 127. D 128. 14 - 5i