Download Algebra 2 Honors Final Exam Review

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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. Find the product (5x − 3)(x 3 − 5x + 2).
a. 5x 4 − 3x 3 − 25x 2 + 25x − 6 b. 5x 3 + 22x 2 − 5x − 6 c. 5x 3 − 28x 2 + 25x − 6 d. 5x 4 − 3x 3 + 25x 2 − 5x − 6
Simplify the given expression.
2.
5(a 2 + 5a + 6)
÷
41(a + 3)
6(a + 6)
3(a − 36)
10(a + 3)(a − 2)
10(a + 2)
10(a + 2)
10(a + 3)(a + 2)
a.
b.
c.
d.
41(a − 6)
41(a − 6)
41(a + 6)
41(a + 6)(a − 6)
2
2
12x 3y
⋅
3.
2y 24x 3
3y 2
3y
4y
3y
a.
b.
c.
d.
2
2
4x
4x
4x
3x 2
4. Which gives the solution(s) of the equation
a. –211, 221 b. 41 c. 221 d. –211
3
x − 5 = −6?
Solve the given equation.
5.
9x − 9 + 5 = 10
104
34
109
14
a.
b.
c.
d.
9
9
9
9
6. 2 9 n − 11 =
1
16
7
5
8
a. n =
b. n =
c. n =
d. n = 7
9
3
9
7. 625 x − 4 = 25 x + 5
a. 14 b. 13 c. –13 d. –3
8. Solve the equation. −x 2 + 4 = 2x 2 − 5
Factor the polynomial completely.
9. 30x 3 − 50x 2 + 27x − 45
a. 10x 2 (3x − 5) − 9(3x − 5) b. (10x 2 + 9)(3x − 5) c. 10x 2 (3x − 5) − 27x + 45 d. (30x 3 − 50x 2 ) + (27x − 45)
10.
8a 4 b 2 − 12a 3 b 2
a. 4(2a 4 b 2 − 3a 3 b 2 ) b. a 3 b 2 (8a − 12) c. 4a 3 b 2 (2a − 3) d. 4a 2 b 2 (2a 2 − 3)
1
Graph:
11.
Which is the graph of f(x) =
3
x + 1 − 5?
a.
c.
b.
d.
ÁÊ 1 ˜ˆ
12. Evaluate the expression log3 ÁÁÁÁ ˜˜˜˜ .
Ë 27 ¯
1
1
a. − b. −3 c.
d. 3
3
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 = −2
b. vertex: (2, –4);
axis of symmetry: x = 2
c. vertex: (–2, –4);
axis of symmetry: x = −2
d. 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. 3x 2 = 147
a. ±21 b. ±
144 c. ± 441 d. ±7
2
16. Graph the rational function f (x ) =
a.
1
. Then find its domain and range.
x− 1
c.
Domain: all real numbers except –1
Range: all real numbers except 1
Domain: all real numbers except 1
Range: all real numbers except 1
b.
d.
Domain: all real numbers except –1
Range: all real numbers except 0
Domain: all real numbers except 1
Range: all real numbers except 0
Simplify the given expression. Assume that no variable equals 0.
Ê
ˆÊ
ˆ
17. ÁÁ 19x −6 y 11 ˜˜ ÁÁ −6xy 5 ˜˜
Ë
¯Ë
¯
a. −114x −5 y 16 b.
13y 16
x5
c.
−114y 16
x5
d. −114x −7 y −24
2
ÁÊÁ 32x 18 y 10 ˜ˆ˜
ÁÁ
˜˜
18. ÁÁ
˜
ÁÁ 16x 9 y 20 ˜˜˜
Ë
¯
a. 2x 9 y 20 b.
4x 18
4x 9
c.
d. 4x 18 y −20
y 20
y 10
Simplify the expression.
245
64
19.
a.
7 5
b.
8
49
c.
8
5
7 7
d.
8
8
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 − 12i 2 c. 24 + 0i d. 12 + 0i + 12
3
22. Divide x 3 + x 2 − x + 2 by x + 4.
a. x 2 − 3x + 11, R –42 b. x 2 − 3x + 11 c. x 2 + 5x − 13 d. x 2 + 5x − 13, R 46
23. Divide
a.
xy 8
5
5x 3
25
÷ 9 . Assume that all expressions are defined.
2
3x y 3y
b.
125x
x
5
c.
d.
10
8
9y
5y
xy 8
24. Suppose that x and y vary inversely, and x = 2 when y = 8. Write the function that models the inverse variation.
16
10
6
b. y =
c. y = 4x d. y =
a. y =
x
x
x
25. Solve 8 x + 8 = 32 x .
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 ≥ 3
range: all real numbers
b. maximum value: –3
domain: all real numbers ≤ −3
range: all real numbers
c. maximum value: 3
domain: all real numbers
range: all real numbers ≤ 3
d. minimum value: –3
domain: all real numbers
range: 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 − 2
x+2
x−2
c. y =
d. y =
4
4
4
What are the solutions?
28. 9x 2 + 16 = 0
4 4
16 16
3 3
4 4
a. − i, i b. − i, i c. − i, i d. − ,
3 3
9 9
4 4
3 3
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 = x 2 − 4x − 15 in standard form.
a. y + 4 = (x − 2 )
2
b. y + 19 = (x − 2 )
2
c. y − 2 = (x − 2 )
4
2
d. y + 15 = (x − 2 )
2
31. Consider the function f(x) = −4x 2 − 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.
c. The parabola opens upward.
The axis of symmetry is the line x = −1.
The axis of symmetry is the line x = −1 .
The vertex is the point (−1,14).
The vertex is the point (−1,−6).
The y-intercept is 10.
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.
33.
17 + 7 5
17 + 7 5
17 − 7 5
17 − 7 5
b.
c.
d.
11
11
2
2
192 − 245 + 27 + 80
a. 11 3 − 3 5 b. 11 3 + 3 5 c. 3 3 − 11 5 d. 3 3 + 11 5
5
Divide.
Ê
ˆ
34. ÁÁ 2x 4 − 4x 3 − 12x − 15 ˜˜ ÷ (x − 3 )
Ë
¯
3
3
a. 2x 2 + 2x + 6 +
b. 2x 2 + 2x + 6 + 3 c. 2x 3 + 2x 2 + 6x + 6 + 3 d. 2x 3 + 2x 2 + 6x + 6 +
x−3
x−3
35. Tell whether the function y = 2 (5) shows growth or decay. Then graph the function.
a. This is an exponential growth function.
c. This is an exponential decay function.
x
b.
This is an exponential growth function.
d.
This is an exponential growth function.
36. 16x 2 − 25
a. (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
3
=
k
+1
k −1
a. 3 b. 0 c. 2 d. 1
2
6
Find the sum or difference.
Ê
ˆ Ê
ˆ
38. ÁÁ 6q 5 + 8q 2 + 3 ˜˜ + ÁÁ 8q 5 − 3q − 7 ˜˜
Ë
¯ Ë
¯
5
2
a. −2q + 8q − 3q + 10 b. 14q 5 + 8q 2 − 3q − 4 c. −2q 5 + 8q 2 + 3q + 10 d. 14q 5 + 5q 2 − 4
Solve the system by graphing.
ÏÔÔ
ÔÔ −2x = 3y − 2
39. ÌÔ
ÔÔ x − y = −4
Ó
a.
c.
(–2, –2)
(2, –2)
b.
d.
(–2, –2)
(–2, 2)
What is the quotient in simplified form? State any restrictions on the variable.
40.
x 2 − 16
x 2 + 5x + 4
÷
x 2 + 5x + 6 x 2 − 2x − 8
(x − 4) 2
(x + 4) 2 (x + 1)
(x − 4) 2
1
a.
b.
c.
d.
2
(x + 3)(x + 1)
(x + 2) (x + 3)
(x + 3)(x + 1)
(x + 3)(x + 1)
7
Solve the equation.
41. 5n − 2 (n − 2) = −11
42.
1Ê
Á 3y + 2 ˜ˆ = 7
¯
4Ë
43. 3x + 17 − 5x = 12 − (6x + 3)
44. –3x + 25 + x + 21 = 2
a. 22 b. –3 c. –22 d. 3
What is the graph of the rational function?
45. y =
−3x + 5
−5x + 2
a.
b.
c.
d.
Solve the equation.
46. x 3 + 4x 2 − 25x − 100 = 0
a. −4, 25 b. −5, 4, 5 c. 4, 5 d. −4, − 5, 5
47. x 3 − x 2 − 6x = 0
8
Solve the equation. Check for extraneous solutions.
48. log 5 (3x + 9) = 2
23
16
1
34
a.
b.
c.
d.
3
3
3
3
49. Add. Write your answer in standard form.
(4d 5 − d 3 ) + (d 5 + 6d 3 − 4)
a. 5d 5 + 5d 3 − 4 b. 5d 5 + 5d 3 c. 5d 10 + 5d 6 − 4 d. 4d 5 + 6d 3 − 4
50. Which is the graph of the function f(x) = x 3 − 2x ?
a.
b.
c.
d.
x+6
−12x − 59
+
.
x − 7 x 2 − 3x − 28
x+5
−11x − 53
x 2 + 10x + 24
x+6
a. 2
b.
c.
d.
(x
+
4)(x
−
7)
(x
−
7)(x
+
4)
x
+4
x − 2x − 35
51. Add
52. Express log 2 64 − log 2 4 as a single logarithm. Simplify, if possible.
a. log2 4 b. 8 c. 4 d. log 2 60
9
What is the expression in factored form?
53. 16x 2 + 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) = 4x 2 − 8x + 3 .
a. x = −1 b. y = −1 c. y = 1 d. x = 1
Solve the linear system.
55. −4x − 3y = −27
−4x + 4y = 8
a. (–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 ) = 2x 6 − 10x 5 − 23x 4 + 80x 3 + 28x 2 − 20x + 9
1 3 9
1 3 9
1 3 9
1 3 9
a. ±1, ±3, ±9, ± , ± , ± b. 1, 3, 9, ± , ± , ± c. 1, 3, 9, , ,
d. –1, –3, –9, − , − , −
2 2 2
2 2 2
2 2 2
2 2 2
Determine the solution of the system of inequalities.
57. y ≤ −x − 1
−2x + y ≥ −2
a.
c.
b.
d.
10
Factor completely.
58. 60z 6 − 118z 5 + 56z 4
a. 2z 4 (5z − 7)(6z − 4) b. z 4 (6z − 7)(5z − 4) c. z 4 (5z + 7)(6z − 4) d. 2z 4 (6z − 7)(5z − 4)
59. Find the y-intercept of the equation. y = − 3 ⋅ 7 x
a. 4 b. –21 c. –3 d. 7
Solve the following system of equations by graphing.
60. 2y + 8x = 58
y − 5x = 11
a. (2, 21) b. (21, 2) c. (4, 20) d. (1, 21)
61. Find the zeros of f(x) = x 2 + 7x + 9 by using the Quadratic Formula.
a. x =
−7 ± 13
2
b. x = −7 ±
62. Simplify the expression
a. 4 256 z 4 b. 4z 4 c.
4
13 c. x =
3±
7
2
d. x = 3 ±
7
256z 16 . Assume that all variables are positive.
4
256 z 11 d. 4z 11
63. Which represents the graph of y =
a.
x + 4 ? State the domain and range of the function.
c.
Domain: x ≥ −4; Range: y ≥ 0
b.
Domain: x ≥ 4; Range: y ≥ 0
d.
Domain: x ≥ 0; Range: y ≥ −4
Domain: x ≥ 0; Range: y ≥ 4
11
Solve the given equation. If necessary, round to four decimal places.
64. 9 2x = 21
a. 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 x 4 − 3x 3 − x 2 − 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.
x 2 + 9x + 18
÷
x2 − 9
9x + 6
a.
b.
3
x+6
x−6
x−6
x−9
x+3
c.
d.
x−3
x−3
x−6
68. Simplify log 7 x 3 − log 7 x .
a. log 7 (x 3 − x) b. 2log 7 x c. log 7 2x d. 2(x 3 − x)
12
Solve the equation by factoring.
69. 2x 2 + 3x − 14 = 0
7
7
a. {–4, − } b. {− , 2} c. {–4, 7} d. {2, 7}
2
2
Solve the system of inequalities by graphing.
ÔÏÔÔ y ≤ −3x − 1
70. ÔÔÌ
ÔÔ y > 3x − 2
Ó
a.
b.
d.
c.
71. Multiply
8x 4 y 2 9xy 2 z 6
⋅
. Assume that all expressions are defined.
3z 3
4y 4
a. 6x 4 yz 2 b. 6x 5 y 8 z 9 c. 6x 5 z 3 d.
3
2
x3y2z
6x
4x + 6
=
.
x−3
x−3
3
a. There is no solution. b. x = − 2 c. x = 3 d. x = −3
72. Solve the equation
73. Which is equivalent to 81 −1/4 ?
a. 9 b. 3 c.
1
1
d.
9
3
74. Which gives the solution(s) of x + 72 = x?
a. –8 b. 9 c. no solution d. 9, –8
13
75. Find the zeros of the function h (x ) = x 2 + 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
b.
–3, 3, 4
c.
3, –3, 4
d.
–3, 3, –4
77. Solve the equation 2x 2 + 18 = 0.
a. x = ±3i b. x = ±3 c. x = 3 ± i d. x = ±3 + i
ÏÔÔ
ÔÔ 3x + y = −3
78. Use substitution to solve the system ÌÔ
.
ÔÔ y = x + 5
Ó
a. (− 3 , –3) b. (−2, 3) c. (− 3 , 1) d. (3, −2)
8
4
79. Find the value of f(–9) and g(–2) if f (x ) = −5x − 2 and g (x ) = 3x 2 − 21x .
a. f(–9) = –7
c. f(–9) = 43
g(–2) = 19
g(–2) = 54
b. f(–9) = 47
d. f(–9) = 10
g(–2) = –83
g(–2) = –27
Determine the value or values of the variable where the expression is not defined.
80.
x−7
x 2 − 6x − 16
14
What is the solution of each equation?
81. 108x 2 = 147
49 49
7 7
6 6
36 36
a. − ,
b. − ,
c. − ,
d. − ,
36 36
6 6
7 7
49 49
10 − x 2 − 3x
. Identify any x-values for which the expression is undefined.
x 2 + 2x − 8
−x − 5
x+5
c.
x+4
x+4
The expression is undefined at x = −4.
The expression is undefined at x = 2 and
x = −4 .
−x − 5
x+5
d.
x+4
x+4
The expression is undefined at x = 2 and
The expression is undefined at x = −4 .
x = −4.
82. Simplify
a.
b.
83. Determine whether the binomial (x − 4 ) is a factor of the polynomial P (x ) = 5x 3 − 20x 2 − 5x + 20 .
a. (x − 4 ) is not a factor of the polynomial P (x ) = 5x 3 − 20x 2 − 5x + 20 . b. (x − 4 ) is a factor of the
polynomial P (x ) = 5x 3 − 20x 2 − 5x + 20 . c. Cannot determine.
84. Express as a single logarithm: log a 13 + log a 60
ÊÁ 13 ˆ˜
a. log a 780 b. log a (13+ 60) c. log a ÁÁÁÁ ˜˜˜˜ d. log a 13 60
Ë 60 ¯
ÔÏÔÔ y < −3x + 2
Ô
85. Graph the system of inequalities ÌÔ
.
ÔÔ y ≥ 4x − 1
Ó
a.
b.
c.
d.
15
86. Solve using factoring: 3x 2 + 5x − 12 = 0
4
4
a. − , 3 b. 4, − 3 c. 8, − 18 d. , − 3
3
3
x 2 + x − 30
= 11. Check your answer.
x−5
a. x = 5 b. x = 16 c. x = −6 d. There is no solution because the original equation is undefined at x = 5.
87. Solve
Perform the indicated operation(s) and simplify.
88.
4
1
+
x+8 x−8
5
5x− 24
5
5x− 24
a.
b. 2
c.
d. 2
x+8
5
x − 64
x − 64
−6x 2 + x − 3 −2x − 4
− 2
. Identify any x-values for which the expression is undefined.
x +9
x2 + 9
−6x 2 − x − 7
−6x 2 − x − 7
a.
; The expression is always defined. b.
; The expression is undefined at x = ±3.
2
x +9
x2 + 9
−6x 2 + 3x + 1
−6x 2 + 3x + 1
;
The
expression
is
undefined
at
x
=
±3.
d.
;
c.
x2 + 9
x2 + 9
The expression is always defined.
89. Subtract
90. Which is the graph of f (x ) =
x−2
?
x−1
a.
c.
b.
d.
91. Divide by using synthetic division.
(x 2 − 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 +
16
10
x−2
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:
ÊÁ 1 ˆ˜ x
93. f(x) = 2ÁÁÁÁ ˜˜˜˜
Ë 4¯
a.
c.
b.
d.
1
3
2
3
94. Simplify the expression (27) ⋅ (27) .
a. 9 b. 3 c. 27 d. 729
What are the solutions of the following systems?
ÏÔÔ
ÔÔ −x + 2y = 10
95. ÌÔ
ÔÔ −3x + 6y = 11
Ó
a. infinitely many solutions b. (–5, 2) c. (5, –2) d. no solutions
Simplify:
96.
−3x + 3x 2
−24x + 24
x
x − x2
x2
1−x
a. −
b.
c.
d.
8
8x − 8
16
16
Find the inverse of the given function.
97. f (x ) =
a. f
−1
7x − 3
16
(x ) =
16x − 3
b. f
7
−1
(x ) =
16x + 3
c. f
7
−1
17
(x ) =
7x + 16
d. f
3
−1
(x ) =
7x − 16
3
ÔÏÔÔ −5x + 4y = 6
Ô
98. Use a graph to solve the system ÌÔ
. Check your answer.
ÔÔ 3x − y = 2
Ó
a.
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).
The solution to the system is (2, –4).
Simplify the rational expression, if possible.
99.
n 2 + 2n − 24
n 2 − 11n + 28
n+6
n+6
n+6
n−4
a.
b.
c.
d.
n+7
n−7
n−4
n−7
100. Write a quadratic function in standard form with zeros 6 and –8.
a. f(x) = x 2 + 2x − 48 b. f(x) = x 2 − 2x − 48 c. f(x) = x 2 − 4x + 4 d. 0 = x 2 + 2x − 48
101. Let f(x) = x 2 − 5 and g(x) = 3x 2 . Find g(f(x)) .
a. 3x 4 − 30x 2 + 75 b. 3x 4 − 15 c. 3x 4 − 5 d. 9x 4 − 5
102. Simplify 8 4 / 3 .
a.
1
32
b. 8 c.
d. 16
2
3
Solve.
103. x 2 − 6x = 0
a. 0, 6 b. 0, –6 c. –6, 6 d. 1, 6
18
104. Graph the function. Label the vertex, axis of symmetry, and x-intercepts.
y = 2 (x + 2 ) (x + 4 )
a.
c.
vertex: (−3, −2)
axis of symm: x = −3
x-intercepts: –4, –2
b.
vertex: (3, −2)
axis of symm: x = 3
x-intercepts: 2, 4
d.
vertex: (−3, 2)
axis of symm: x = −3
x-intercepts: –4, –2
vertex: (3, 2)
axis of symm: x = 3
x-intercepts: 2, 4
Find the roots of the polynomial equation.
105. x 3 − 2x 2 + 10x + 136 = 0
a. –3 ± 5i, –4 b. 3 ± 5i, –4 c. –3 ± i, 4 d. 3 ± i, 4
106. Solve log3 x = 6.
a. 18 b. 216 c. 6 d. 729
Simplify the sum.
107.
w 2 + 2w − 24
+
8
w + w − 30
w−5
2
w−4
w + 2w − 16
w+4
a.
b.
c. w + 4 d.
2
w−5
w + w − 30
w−5
2
108. Factor out the greatest common monomial factor from 21z 3 + 28z.
19
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 log 4 16 = 2 in exponential from.
a. 4 16 = 2 b. 4 2 = 16 c. 2 −4 = 16 d. 2 4 = 16
111. Solve the equation x 2 − 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 ) + 17 = 59
2
a. −7 ±
13 b. 42 ±
14 c. 42 ±
113. 8x 2 + 23 = 823
a. no real-number solution b. ±
13 d. −7 ±
14
800 c. ± 64 d. ±10
114. 3x 2 − 9 = 3
Solve by using tables. Give each answer to at most two decimal places.
115. 2x 2 + 5x − 3 = 0
a. 0.5, –3 b. 1, –6 c. 3, –0.5 d. 1.75, –1.75
116. Solve the polynomial equation 3x 5 + 6x 4 − 72x 3 = 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.
2
3
+
2x
+5
4x − 25
4x + 7
4x − 10
4x − 7
5
a.
b.
c.
d.
2
(2x + 5)(2x − 5)
(2x − 5)(2x + 5)
(2x + 5)(2x − 5)
(4x + 2x − 20)
2
Solve the equation. Check the solution.
118.
g+4
=
g−5
g−2
g−8
22
a. −
b. 22 c. −22 d. 14
3
Solve.
119. x 2 − 10x + 29 = 0
a. −5 + 4i, −5 − 4i b. −5 + 2i, −5 − 2i c. 5 + 4i, 5 − 4i d. 5 + 2i, 5 − 2i
20
Find the quotient.
120.
x + 4 x 2 − 16
÷
x−4
4−x
x+4
1
1
1
a.
b.
c.
d.
x−4
4−x
x−4
2−x
Evaluate each expression.
121. 4
log 4 8.2
a. 8.2 4 b. 8.2 c. 4 d. 4 8.2
122. Simplify the expression log 4 64.
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:
12 +
124.
48
a. 6 3 b.
60 c. 24 3 d. 3 6
Multiply the expressions. Simplify the result.
125.
5
d 2 5e f
·
ef
4d
a.
5de 4
5
4d 3
5d 2 e 5
b.
c.
d.
4
4
4def
5e 6 f 2
126. Is (x − 2) a factor of P(x) = x 3 + 2x 2 − 6x − 4 ? If it is, write P(x) as a product of two factors.
a. yes:
c. yes:
2
P(x) = (x + 2)(x + 4x + 2)
P(x) = (x − 2)(x 2 − 4x + 2)
b. yes:
d. (x − 2) is not a factor of P(x)
2
P(x) = (x − 2)(x + 4x + 2)
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)
21
ID: A
________________________________________________________________
Answer Section
1. A
(5x − 3)(x 3 − 5x + 2)
Distribute 5x and −3.
= 5x(x 3 − 5x + 2) − 3(x 3 − 5x + 2)
= 5x(x 3 ) + 5x(−5x) + 5x (2) − 3(x 3 ) − 3 (−5x) − 3 (2)
Distribute 5x and −3 again.
= 5x − 25x + 10x − 3x + 15x − 6
= 5x 4 − 3x 3 − 25x 2 + 25x − 6
Multiply.
Combine like terms.
4
2
3
2. B
To divide two rational expressions, multiply the first expression by the reciprocal of the divisor.
3. C
First, 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. A
Eliminate 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. C
Find 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. B
Simplify 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
a
=
b
a
b
20. A
Combine the real and imaginary parts of the complex numbers to add them.
1
ID: A
21. C
Use the FOIL method to multiply the complex numbers and use the formula i 2 = −1. Combine the real parts
and then the imaginary parts of the two numbers.
22. A
23. A
5x 3
25
÷ 9
2
3x y 3y
Rewrite as multiplication by the reciprocal.
9
5x 3 3y
= 2 ⋅
3x y 25
Simplify by canceling common factors.
xy 8
=
5
24. A
25. C
26.
27.
28.
29.
x+8
Ê ˆx
ÁÊÁ 2 3 ˆ˜˜
Rewrite each side as powers of the same base.
= ÁÁ 2 5 ˜˜
Ë ¯
Ë ¯
3(x + 8)
To raise a power to a power, multiply the exponents.
= 2 5x
2
3(x + 8) = 5x
The bases are the same, so the exponents must be equal.
x = 12
The solution is x = 12.
D
D
A
C
x
f(x) = 4 + (–5)
The variable, x, is divided by 4, then –5 is added.
Undo the addition by subtracting –5. Undo the division by
f −1 (x) = 4(x – (–5))
multiplying by 4.
−1
f (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 = 14
The 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. A
Find the principal square root of each term of the radicand and simplify the expression.
2
ID: A
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.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
Step 2 Choose several values of x and generate ordered pairs. Then, graph the ordered pairs and connect with
a smooth curve.
C
C
B
D
C
–5
26
3
–2
A
A
D
−2, 0, 3
B
A
(4d 5 − d 3 ) + (d 5 + 6d 3 − 4)
Identify like terms. Rearrange terms to get like terms
= (4d 5 + 6d 3 ) + (−d 3 + d 5 ) + (−4)
together.
Combine like terms.
= 5d 5 + 5d 3 − 4
50. B
51. D
x+6
−12x − 59
+
x − 7 (x + 4)(x − 7)
ÊÁ x + 4 ˆ˜ x + 6
−12x − 59
˜˜
+
= ÁÁÁÁ
˜
˜
Ë x + 4 ¯ x − 7 (x + 4)(x − 7)
=
ÊÁ x + 4 ˆ˜
˜˜ .
Multiply by ÁÁÁÁ
˜˜
x
+
4
Ë
¯
x 2 + 10x + 24
−12x − 59
+
(x + 4)(x − 7) (x + 4)(x − 7)
x 2 − 2x − 35
(x + 4)(x − 7)
(x + 5)(x − 7)
=
(x + 4)(x − 7)
x+5
=
x+4
=
Factor the denominators. The LCD is (x + 4)(x − 7).
Add the numerators.
Factor the numerator.
Divide the common factor.
52. C
To subtract the logarithms divide the numbers.
64
log 2 64 − log 2 4 = log 2 ( 4 ) = log 2 16 = 4
53. C
3
ID: A
54. D
55.
56.
57.
58.
59.
60.
61.
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.
C
A
Use the Rational Zero Theorem.
C
D
C
A
Graph the equations and find their point of intersection.
A
Set f(x) = 0 .
x 2 + 7x + 9 = 0
x=
x=
x=
x=
−b ±
b 2 − 4ac
2a
Write the Quadratic Formula.
−7 ±
(7) 2 − 4(1)(9)
2(1)
Substitute 1 for a, 7 for b, and 9 for c.
−7 ±
49 − 36
2
Simplify.
−7 ± 13
2
Write in simplest form.
62. B
4
256z 16
4
= 256 · z 4 · z 4 · z 4 · z 4
= 4· z· z· z· z
= 4z 4
Factor into perfect powers of four.
Use the Product Property of Roots.
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
4
ID: A
66. B
The 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
±1
p = −90 and q = 1
Step 2: Graph x 4 − 3x 3 − x 2 − 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
1
5
10
45
90
2
9
18
0
Test the possible root of −2:
−2| 1 −3 −1 −27 −90
1
−2
10
−18
90
−5
9
−45
0
The polynomial factors into (x − 5 ) (x + 2 )(x 2 + 9) = 0.
Step 4: Solve x 2 + 9 = 0 to find the remaining roots.
x2 + 9 = 0
x 2 = −9
x = ±3i
The 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
log 7 x 3 − log 7 x
= 3log7 x − log 7 x
Use the Power Property of Logarithms.
= 2log 7 x
Simplify.
69. B
For 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
5
ID: A
71. C
Arrange the expressions so like terms are together:
8 ⋅ 9(x 4 ⋅ x)(y 2 ⋅ y 2 )z 6
3 ⋅ 4 ⋅ z3y4
.
Multiply the numerators and denominators, remembering to add exponents when multiplying:
72x 5 y 4 z 6
12z 3 y 4
Divide, remembering to subtract exponents: 6x 5 y 0 z 3 .
Since y 0 = 1, this expression simplifies to 6x 5 z 3 .
72. A
6x
4x + 6
(x − 3) =
(x − 3)
Multiply each term by the LCD, (x – 3).
x−3
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 ) = x 2 + 23x + 60
Set the function equal to 0.
x 2 + 23x + 60 = 0
(x + 20)(x + 3) = 0
Factor: Find factors of 60 that add to 23.
x + 20 = 0 or x + 3 = 0
Apply the Zero-Product Property.
x = −20 or x = −3
Solve each equation.
76. B
77. A
2x 2 + 18 = 0
2x 2 = −18
x 2 = −9
x = ± −9
x = ±3i
78. B
Step 1
Step 2
Step 3
Step 4
Add −18 to both sides.
Divide both sides by 2.
Take square roots.
Express in terms of i.
y = x+5
3x + y = −3
3x + (x + 5 ) = −3
4x + 5 = −3
4x = −8
4x −8
=
4
4
x = −2
y = x+5
y = −2 +5
y=3
The second equation is solved for y.
(−2, 3)
Write the solution as an ordered pair.
Substitute x + 5 for y in the first equation.
Simplify and solve for x.
Divide both sides by 4.
Write one of the original equations.
Substitute −2 for x.
Find the value of y.
6
.
ID: A
79. C
Substitute 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(x 2 + 3x − 10)
Factor −1 from the numerator and reorder the terms.
x 2 + 2x − 8
−1(x + 5)(x − 2)
=
Factor the numerator and denominator.
(x + 4)(x − 2)
−x − 5
Divide the common factors and simplify.
=
x+4
The expression is undefined at those x-values, 2 and −4, that make the original denominator 0.
83. B
Find P (4) by synthetic substitution.
4
5
5
84.
85.
86.
87.
88.
89.
−20
−5
20
20
0
−20
0
−5
0
Since P (4) = 0, x − 4 is a factor of the polynomial P (x ) = 5x 3 − 20x 2 − 5x + 20 .
A
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.
D
D
x 2 + x − 30
Note that x ≠ 5.
= 11
x−5
(x − 5 ) (x + 6 )
Factor.
= 11
x−5
x + 6 = 11
The factor (x − 5) cancels.
x=5
Because the left side of the original equation is undefined when x = 5, there is no solution.
D
D
−6x 2 + x − 3 −2x − 4
− 2
x2 + 9
x +9
−6x 2 + x − 3 + 2x + 4
Subtract the numerators. Distribute the negative sign.
x2 + 9
−6x 2 + 3x + 1
Combine like terms.
=
x2 + 9
There is no real value of x for which x2 + 9 = 0; the expression is always defined.
90. D
=
7
ID: A
91. C
For (x − 2) , a = 2.
2
1
–9
2
1
–7
10
–14
–4
Write the coefficients of the expression.
Bring down the first coefficient. Multiply and add each
column.
Write the remainder as a fraction to get x − 7 +
−4
x−2
.
92. C
Use inverse proportion to relate values in the following equation.
x2
x1
=
y2
y1
93. D
94. C
1
3
(27) ⋅ (27)
1+2
3
= (27)
= (27) 1
= 27
2
3
Product of Powers
Simplify.
95. D
96. A
97. B
The inverse function can be found by exchanging the domain and range of the function.
98. A
Solve each equation for y.
ÔÏÔÔ
ÔÔ y = 54 x + 32
ÌÔÔ
ÔÔ y = 3x − 2
Ó
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 = 6 or x = −8
Write the zeros as solutions for two equations.
x − 6 = 0 or x + 8 = 0
Rewrite each equation so that it is equal to 0.
Apply the converse of the Zero-Product Property to write a
0 = (x − 6)(x + 8)
product that is equal to 0.
2
Multiply the binomials.
0 = x + 2x − 48
2
Replace 0 with f(x)
f(x) = x + 2x − 48
101.
102.
103.
104.
105.
106.
A
D
A
A
B
D
Use the definition of logarithms with base b.
107. D
8
ID: A
Ê
ˆ
108. 7z ÁÁ 3z 2 + 4 ˜˜
Ë
¯
109. A
A = Pe rt
0.05 (8 )
A = 6000e
A ≈ 8950.95
Substitute 6,000 for P, 0.05 for r, and 8 for t.
Use the [e^x] key on a calculator.
The total amount after 8 years is $8950.95.
110. B
Logarithmic form: log 4 16 = 2
Exponential form: 4 2 = 16
The base of the logarithm becomes the base of the power, and the logarithm is the exponent.
111. C
x 2 − 10x + 25 = 54
2
Factor the perfect square trinomial.
(x − 5 ) = 54
112.
113.
114.
115.
116.
x − 5 = ± 54
x = 5 ± 54
Take the square root of both sides.
x = 5 ±3 6
Simplify.
Add 5 to each side.
D
D
±2
A
B
3x 5 + 6x 4 − 72x 3 = 0
Ê
ˆ
3x 3 ÁÁ x 2 + 2x − 24 ˜˜ = 0
Ë
¯
3
3x (x + 6 ) (x − 4 ) = 0
3x 3 = 0, x + 6 = 0, x − 4 = 0
x = 0, x = −6, x = 4
Factor out the GCF, 3x3.
Factor the quadratic.
Set each factor equal to 0.
Solve for x.
117. C
Find 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 4 3 , and apply the Inverse Properties of Logarithms and Exponents.
123. B
124. A
125. A
126. B
127. D
128. 14 - 5i
9