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Algebra 2E Final Review
Factor the polynomial completely.
1. 18x3 – 30x2 + 60x – 100
2. 216x3 + 125y3
3. Solve polynomial equation by factoring. x4 – 11x2 + 24 = 0
4. Use synthetic substitution to find g(4) and g(–6) for the function g(x) = 8x4 – 2x2 + 10x – 3.
5. Use synthetic substitution to find g(3) and g(–8) for the function g(x) = x5 – 8x3 – 3x + 2.
Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some of the factors may not be
binomials.
6. 64x3 – 16x2 – 175x – 98; x – 2
7. Describe the possible real zeros of f(x) = –7x3 + 8x2 + 4x – 3.
a. 4, 2, or 0 2 or 0 positive zeros and 1 negative zero
b. 4, 2, or 0 positive zeros and 0 negative zeros
c. 4, 2, or 0 2 or 0 positive zeros and 1 negative zero
d. 4, 2, or 0 2 or 0 positive zeros and 0 negative zeros
8. List all of the possible rational zeros of the following function.
f(x) = x6 – 4x5 – 17x4 + 90x3 + 28x2 – 22x + 100
9. Describe the end behavior of the graph.
10. Describe the end behavior of the graph and state the number of real zeros.
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Algebra 2E Final Review
11. Describe the end behavior of the graph and state the number of real zeros.
12. Find
for the following functions.
f(x) = 2x2 + 3x + 2
g(x) = 8x + 2
13. Find
for the following functions.
f(x) = –8x3 + 20x2 – 5
g(x) = 11x2 + 23
14. Find
for the following functions.
2
f(x) = 7x – 4x – 7
g(x) = 7x – 7
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Algebra 2E Final Review
15. Find
for the following functions.
16. Find
and
.
g(x) = 6x
h(x) = –9x3 + 11x2 – 7x + 3
17. Find
g(x) = 11x
h(x) = –7x – 6
and
.
Find the inverse of the given function.
18. f(x) =
x – 11
19. f(x) =
20. Graph the given function. State the domain and range.
21. Graph the inequality
.
22. Graph the inequality
.
Simplify.
23.
Simplify.
24.
+
25.
+
26. (3 +
27. (
–
–
)(5 +
–
+
)
)2
28.
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Algebra 2E Final Review
29.
Solve the given equation.
30.
Sketch the graph of the given function. Then state the function’s domain and range.
31.
32. y = –2.5(4)x
Evaluate the logarithmic expression.
33. log8 64
34. log2
Solve the given equation.
35. 37n – 6 =
36. 1011n + 10 = 10,000
Evaluate the logarithmic expression.
37. Graph f(x) =
38. Graph f(x) =
.
.
39. Solve log27 n = .
40. Solve log5 x = 6.
Solve the given equation. If necessary, round to four decimal places.
41. log2 5 + log2 a = log2 19
42. log5 (x + 2) – log5 6 = log5 36
43. 13y = 50
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44. 93r = 28
Express the given logarithm in terms of common logarithms. Then approximate its value to four decimal places.
45. log4 5.4
46. log4 17
Solve the given equation. Round to the nearest ten-thousandth, if necessary.
47. 8 + 3e4x = 27
48. 5ex – 8 = 9
Solve for x.
49.
50.
51.
52. log4 x = 6
53.
Solve each equation.
54.
55.
56. 2
57. Find the inverse of the function. Then graph the function and its inverse.
58. The height of a conical container is equal to its radius. The volume of a cone with equal height and radius can be
written as
. If the volume of the cone is 600 cubic centimeters, find its radius and height. Round the answer
to the second decimal place. Use
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.
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Algebra 2E Final Review
59. The sale of a certain product is given by the equation
, where s is the total revenue of sold goods and q is
the number of goods sold. Write the equation without a radical in the denominator.
60. The winds at higher altitudes generally have a higher velocity than the winds at ground level. In other words, at any
given time and place, wind speed usually increases with altitude. It is modeled in the formula
, where uz =
wind velocity at height z , ug = wind velocity at ground station height , hz = height z , hg = ground station height, and n = a
function of the Pasquill stability class and the terrain type.
Given a wind speed of 5 meters per second measured at 10 meters above the ground and n = , calculate the wind speed at
160 meters above ground.
Simplify the given expression.
61.
62.
63.
64.
Simplify the given expression.
65.
+
Determine the value(s) of x for which the function is not defined.
66. f(x) =
Identify the asymptotes of each function.
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67.
68. If y varies directly as x and y = 33 when x = –11, find y when x = 34.
69. Suppose y varies jointly as x and z. Find y when x = 15 and z = 13, if y = 198 when x = 6 and z = 11. Round your
answer to the nearest hundredth, if necessary.
70. If y varies inversely as x and y = 196 when x = –19, find y when x = 2. Round your answer to the nearest hundredth, if
necessary.
71. Suppose f varies directly as g, and f varies inversely as h. Find g when f = –8 and h = 7, if g = 81 when h = –3 and f = –
9. Round your answer to the nearest hundredth, if necessary.
72. Suppose f varies directly as g, and f varies inversely as h. Find g when f = 11 and h = –5, if g = –189 when h = 7 and f
= –9. Round your answer to the nearest hundredth, if necessary.
Solve the given equation. Check your solution.
73.
+
=
74. Patrick is constructing a model of a building. The length of the windows in the building can be modeled by the
expression
, and the width of the windows can be modeled by
Write an expression for the perimeter of a
window in Patrick’s model.
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75. Rectangle ADFI has an area of
square meters and a length of
meters. Parallelogram BCHG has an
area of
square meters and a height of
meters. Find the area of the triangle ABC.
76. A raffle prize of
dollars is to be divided among 7x people. Write an expression for the amount of money that
each person will receive.
77. Write an exponential function for the graph that passes through the given points.
3
4
a) (0, ) and (2, 36.75)
b) (0,15) and (2,
15
16
)
78. Simplify.
a)
b)
79. Factor and find the zeros of the polynomial function f(x) = −x4 + 2x2 – 1 using Descartes’ Rule of Signs, The
Remainder Theorem, and the Factor Theorem. Then, sketch the graph of the function using the roots, your knowledge of
end behavior, and other points you may need to establish accuracy (relative minimum / relative maximum).
80. Find (f + g)(x), (f –g)(x), (f ∙ g)(x), and (
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f
g
) (x) for:
f(x) = x2 −1; g(x) =
1
𝑥+1
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Algebra 2E Final Review
Answer Key
1. (6x2 + 20)(3x – 5)
2. (6x + 5y)(36x2 – 30xy + 25y2)
,–
3.
,–
,
4. 2,053, 10,233
5. 20, –28,646
6. (8x + 7)(8x + 7)
7. c
8.
,
,
,
,
,
,
,
9.
10.
5
11.
4
12. 2x2 + 11x + 4
13. –8x3 + 9x2 – 28
14. 49x3 – 77x2 – 21x + 49
15.
,
16.
= –54x3 + 66x2 – 42x + 18
= –1944x3 + 396x2 – 42x + 3
17.
= –77x – 66
= –77x – 6
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Algebra 2E Final Review
18. f–1(x) =
19. f–1(x) =
20.
The domain is x ≤ and the range is y ≤ 4.
21.
22.
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Algebra 2E Final Review
23.
24. 7
25. 5
+ 11
26. 15 + 3
+5
+
27. 15 – 2
28.
29.
30.
31.
The domain is all real numbers and the range is all positive numbers.
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Algebra 2E Final Review
32.
The domain is all real numbers and the range is all negative numbers.
33. 2
34. –3
35. n =
36. n =
37.
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Algebra 2E Final Review
38.
39. 3
40. 15,625
41. 3.8
42. 214
43. 1.5252
44. 0.5055
45.
; 1.2165
46.
; 2.0437
47. 0.4615
48. 1.2238
49. 16
50. –3
51. 8
52. 4096
53. x = 1, x = 5
54.
55.
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Algebra 2E Final Review
56.
57.
58. r = 8.31 cm; Because the height is equal to the radius, the height has the same measure.
59.
or
60.
= 10 m/s
61.
62.
63.
64.
65.
66.
67. x = 3, f(x) = 0
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Algebra 2E Final Review
68. –102
69. 585
70. –1,862
71. –168
72. –165
73. 10
74.
75.
76.
dollars
1
77. a) y = .75( 7)x
b) y= 15( 4 )x
78. a) 96c2d4
b) 12x2 |y|
79.
80.
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