Download Algebra III - Semester Exam Review

Algebra III - Semester Exam Review
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the real solutions of the equation by factoring.
1) x3 - 49x = 0
A) {0, 49}
2) 5x5 = 45x3
A) {0}
C) {-3 5, 0, 3
1)
B) {0, 7, -7}
C) {0, 7}
D) {0, -7}
2)
B) {-3, 3}
D) {-3, 0, 3}
5}
3) x3 + 5x2 - 4x - 20 = 0
A) {2, -5}
3)
B) {-2, 2, -5}
C) {4, -5}
4) x3 + 4x2 + 4x + 16 = 0
A) {-4}
C) {4}
D) {-2, 2, 5}
4)
B) {-2, 2, -4}
D) no real solution
5) 5x4 - 320x2 = 0
A) {0}
C) {-8, 0, 8}
5)
B) {-8 5, 0, 8
D) {-8, 8}
5}
6) 3x4 = 24x
A) {0, 3, 2}
B) {0}
C) {-2, 0, 2}
D) {0, 2}
7) x3 + 4x2 - 12x = 0
A) {0, 2, -6}
B) {0, -2, 6}
C) {-2, 6}
D) {2, -6}
8) x3 + 6x2 - x - 6 = 0
A) {1, -6, 6}
B) {36}
C) {-6, 6}
D) {-1, 1, -6}
6)
7)
8)
9) 10x3 + 70x2 + 120x = 0
9)
B) {- 1 , -3}
4
A) {0, -4, -3}
C) {0, 4, 3}
D) {-4, -3}
Solve the equation by completing the square.
10) x2 + 12x + 21 = 0
A) {-12 + 21}
C) {6 - 21, 6 +
10)
B) {6 + 15}
D) {-6 - 15, -6 +
21}
1
15}
Find the real solutions, if any, of the equation. Use the quadratic formula.
11) 5x2 + x - 1 = 0
11)
A) { 1 - 21 , 1 + 21 }
10
10
B) { -1 - 21 , -1 + 21 }
10
10
C) { -1 - 21 , -1 + 21 }
2
2
D) no real solution
Solve the inequality. Express your answer using interval notation.
12) x(4x - 1) ≤ (2x + 4)2
-5
-4
A) [
-3
-2
-1
0
1
2
3
4
16
, ∞)
17
-5
C) [-
-4
B) (-∞, -
-3
-2
-1
0
1
2
3
4
-5
16
, ∞)
17
-5
-4
12)
-4
D) (-∞, -
-3
-2
-1
0
1
2
3
4
16
]
17
-5
-3
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
16
]
17
-4
-3
13) 17 ≤ 9 x + 8 < 44
2
13)
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1011121314
A) (2, 8]
B) [2, 3)
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1011121314
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1011121314
C) [2, 8)
D) (2, 3]
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1011121314
14)
-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 1011121314
x ≥5 + x
2
12
14)
-16 -12 -8
-4
0
4
8
12
16
20
A) (12, ∞)
-16 -12 -8
B) (-∞, 12]
-4
0
4
8
12
16
20
-16 -12 -8
C) [-12, ∞)
-16 -12 -8
-4
0
4
8
12
16
20
-4
0
4
8
12
16
20
D) [12, ∞)
-4
0
4
8
12
16
20
-16 -12 -8
2
15) -4(6x + 1) < -28x - 20
15)
A) (-∞, 6]
0
1
2
3
4
5
6
7
8
9
10 11 12
-10 -9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
B) (-∞, -4)
C) (-∞, -4]
-10 -9
D) (-4, ∞)
-10 -9
16) 5 - 3(1 - x) ≤ -19
-8
-6
-4
-2
16)
0
2
4
6
8
10
A) (-∞, -7)
-8
B) [-7, ∞)
-6
-4
-2
0
2
4
6
8
10
-8
C) (-∞, -6]
-8
-6
-4
-2
0
2
4
6
8
10
-6
-4
-2
0
2
4
6
8
10
D) (-∞, -7]
-6
-4
-2
0
2
4
6
8
10
-8
17) |x - 7| < 0
A) (-7, 7)
B) (-7, ∞)
C) (-∞, 7)
D) no solution
17)
18) |x + 1| ≤ 0
A) {1}
B) (-∞, -1)
C) {-1}
D) no solution
18)
19) |5x + 2| > 3
A) [-1, 1 ]
5
19)
B) (-1, 1 )
5
C) (-∞, -1) or ( 1 , ∞)
5
D) (-∞, -1] or [ 1 , ∞)
5
20) |x - 2| + 3 ≤ 8
A) [3, 8]
C) (-∞, -3) or (7, ∞)
20)
B) [-3, 7]
D) no solution
3
Solve the problem.
21) A chemist needs 180 milliliters of a 52% solution but has only 44% and 62% solutions available.
Find how many milliliters of each that should be mixed to get the desired solution.
A) 100 mL of 44%; 80 mL of 62%
B) 80 mL of 44%; 100 mL of 62%
C) 70 mL of 44%; 110 mL of 62%
D) 110 mL of 44%; 70 mL of 62%
21)
22) How many gallons of a 30% alcohol solution must be mixed with 60 gallons of a 14% solution to
obtain a solution that is 20% alcohol?
A) 7 gal
B) 12 gal
C) 36 gal
D) 27 gal
22)
23) How many liters of 80% hydrochloric acid must be mixed with 40% hydrochloric acid to get 15
liters of 65% hydrochloric acid? Write your answer rounded to three decimals.
A) 8 L
B) 3.125 L
C) 9.375 L
D) 4.688 L
23)
Use synthetic division to find the quotient and the remainder.
24) x2 + 11x + 16 is divided by x + 8
A) x + 4; remainder 0
C) x + 3; remainder 8
24)
B) x + 3; remainder 0
D) x + 3; remainder -8
25) 6x5 - 5x4 + x - 4 is divided by x + 1
2
25)
A) 6x4 - 8x3 + 5; remainder - 13
2
B) 6x4 - 2x3 - x2 + 1 x + 5 ; remainder - 27
2
4
8
C) 6x4 - 2x3 + x2 - 1 x + 5 ; remainder - 37
2
4
8
D) 6x4 - 8x3 + 4x2 - 2x + 2; remainder -5
Find the vertical asymptotes of the rational function.
26) g(x) = x + 7
x2 + 36
26)
A) x = -6, x = 6
C) x = -6, x = 6, x = -7
27) f(x) =
B) none
D) x = -6, x = -7
x(x - 1)
25x2 + 40x + 7
27)
A) x = - 1 , x = - 7
5
5
B) x = - 1 , x = - 7
25
25
C) x = - 7 , x = 14
25
25
D) x = 1 , x = 7
5
5
2
28) f(x) = -x + 16
x2 + 5x + 4
A) x = -1, x = -4
28)
B) x = -1
C) x = -1, x = 4
4
D) x = 1, x = -4
Form a polynomial whose zeros and degree are given.
29) Zeros: -5, -3, 3; degree 3
A) f(x) = x3 + 9x - 5x2 - 45 for a = 1
29)
B) f(x) = x3 + 9x + 5x2 + 45 for a = 1
D) f(x) = x3 - 9x - 5x2 + 45 for a = 1
C) f(x) = x3 - 9x + 5x2 - 45 for a = 1
30) Zeros: -5, -3, -1, 2; degree 4
A) x4 + 7x3 + 5x2 - 30x - 30
30)
B) x4 + 13x2 - 30
D) x4 - 7x3 + 5x2 + 31x - 30
C) x4 + 7x3 + 5x2 - 31x - 30
For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis
at each x -intercept.
31) f(x) = 2(x - 2)(x - 1)4
31)
A) 2, multiplicity 1, crosses x-axis; 1, multiplicity 4, touches x-axis
B) 2, multiplicity 1, touches x-axis; 1, multiplicity 4, crosses x-axis
C) -2, multiplicity 1, touches x-axis; -1, multiplicity 4, crosses x-axis
D) -2, multiplicity 1, crosses x-axis; -1, multiplicity 4, touches x-axis
32) f(x) = 2(x2 + 1)(x - 4)2
A) 4, multiplicity 2, touches x-axis
B) -1, multiplicity 1, touches x-axis; 4, multiplicity 2, crosses x-axis
C) 4, multiplicity 2, crosses x-axis
D) -1, multiplicity 1, crosses x-axis; 4, multiplicity 2, touches x-axis
32)
Solve the inequality.
33) x2 - 7x ≥ 0
33)
A) (-∞, 0] or [7, ∞)
C) [0, 7]
B) (-∞, -7] or [0, ∞)
D) [-7, 0]
34) x2 - 4x - 5 ≤ 0
A) [5, ∞)
C) (-∞, -1] or [5, ∞)
34)
B) [-1, 5]
D) (-∞, -1]
35) x3 + 5x2 - 24x > 0
A) (-8, ∞)
C) (-8, 0) or (3, ∞)
35)
B) (-∞,-8) or (0, 3)
D) (-3, 0) or (8, ∞)
Use the Remainder Theorem to find the remainder when f(x) is divided by x - c.
36) f(x) = x4 + 8x3 + 12x2; x + 1
A) R = 21
37) f(x) = 5x6 - 3x3 + 8; x + 1
A) R = 16
B) R = 5
C) R = -5
36)
D) R = -21
37)
B) R = 10
C) R = 6
Use the Factor Theorem to determine whether x - c is a factor of f(x).
38) f(x) = x 3 + 3x2 - 8x + 10; x + 5
A) Yes
B) No
5
D) R = 8
38)
39) f(x) = x 3 + 5x2 - 12x + 14; x - 7
A) Yes
B) No
40) f(x) = x 4 - 5x2 - 36; x - 3
A) Yes
B) No
39)
40)
List the potential rational zeros of the polynomial function. Do not find the zeros.
41) f(x) = -4x4 + 3x2 - 4x + 6
41)
A) ± 1 , ± 1 , ± 1 , ± 2 , ± 4 , ± 1, ± 2, ± 4
6
2
3
3
3
B) ± 1 , ± 1 , ± 2 , ± 3 , ± 3 , ± 1, ± 2, ± 3, ± 6
4
2
3
4
2
C) ± 1 , ± 1 , ± 3 , ± 3 , ± 1, ± 2, ± 3, ± 6
4
2
4
2
D) ± 1 , ± 1 , ± 3 , ± 3 , ± 1, ± 2, ± 3, ± 4, ± 6
4
2
4
2
Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the
real numbers.
42) f(x) = 5x3 - 9x2 - 6x + 8
42)
A) 1, 5 , -2; f(x) = (5x - 4)(x - 1)(x + 2)
4
B) -1, 4 , 2; f(x) = (5x - 4)(x - 2)(x + 1)
5
C) -1, 5 , -2; f(x) = (5x - 4)(x - 2)(x + 1)
4
D) -2, 4 , 1; f(x) = (5x - 4)(x - 1)(x + 2)
5
For the given functions f and g, find the requested composite function.
43) f(x) = 5x + 6, g(x) = 5x - 1;
Find (f ∘ g)(x).
A) 25x + 5
B) 25x + 11
C) 25x + 29
44) f(x) = -4x + 9, g(x) = 5x + 9;
Find (g ∘ f)(x).
A) -20x + 54
B) -20x - 36
45) f(x) =
A)
2 , g(x) = 1 ;
x+3
3x
6x
1 - 9x
46) f(x) = x - 7 , g(x) = 2x + 7;
2
A) x + 14
43)
D) 25x + 1
44)
C) 20x + 54
D) -20x + 45
Find (f ∘ g)(x).
B)
45)
2x
1 + 9x
C)
6x
1 + 9x
D)
1x + 3
6x
Find (g ∘ f)(x).
46)
B) 2x + 7
The function f is one-to-one. Find its inverse.
47) f(x) = 8x + 4
A) f-1(x) = x - 4
B) f-1(x) = x + 4
8
8
7
2
C) x
D) x -
C) f-1(x) = x + 4
8
D) f-1(x) = x - 4
8
47)
6
48) f(x) = 4
x
48)
A) f-1(x) = x
4
B) f-1(x) = 4
x
C) f-1(x) = -4x
49) f(x) = x2 + 1, x ≥ 0
A) f-1(x) = x + 1, x ≥ -1
C) f-1(x) =
D) f-1(x) = 4x
49)
B) f-1(x) =
D) f-1(x) =
x - 1, x ≥ 1
x - 1, x ≥ 0
x - 1, x < 0
50) f(x) = 6x2 + 3, x ≥ 0
50)
6
x- 3
A) f-1(x) =
C) f-1(x) = -
x- 3
6
B) f-1(x) =
x- 3
6
D) f-1(x) =
6
x- 3
51) f(x) = x 3 + 4
A) f-1(x) =
C) f-1(x) =
51)
3
3
x+4
B) f-1(x) =
x- 4
D) f-1(x) =
3
3
x+4
x- 4
52) f(x) = 4x - 7
5
A) f-1(x) =
52)
5
4x - 7
B) f-1(x) =
C) f-1(x) = 5x - 7
4
5
4x + 7
D) f-1(x) = 5x + 7
4
Solve the equation.
53) 2(3x - 5 ) = 16
A)
54)
1
8
B) {8}
C) {-3}
D) {3}
1 x
= 1296
6
1
A) 4
55)
53)
54)
B)
1
4
C) {4}
D) {-4}
9 x
= 4
2
81
A) {2}
55)
B)
1
2
1
C) 2
7
D) {-2}
56) e4x - 1 = (e3)-x
56)
A) 1
B)
x=4
3
A) {81}
4
5
C) {0}
D)
1
7
57) log
57)
B) {12}
(x + 4) = 2
5
A) {21}
C) {1.26}
D) {64}
58) log
58)
59) log (x + 2) = log (5x - 3)
5
A) 4
B) {28}
C) {29}
5
3
C)
5
4
C)
5
2
D) {36}
59)
B)
1
D) 4
60) log (2 + x) - log (x - 4) = log 3
A) {7}
60)
B) {-7}
D) ∅
61) log (5x) = log 4 + log (x - 1)
A) - 4
62) log
3
61)
B)
(7x + 8) = log
3
3
4
4
C) 9
D) 4
(7x + 2)
A) {3}
62)
B)
63) log3 x + log3(x - 24) = 4
A) {53}
5
3
C) {0}
D) ∅
C) {-3, 27}
D) ∅
63)
B) {27}
64) 2 + log3(2x + 5) - log3 x = 4
A)
65)
1+
46
9
64)
B)
5
7
C)
1±
46
9
D)
5
4
1 log (x + 6) = log (3x)
2
8
3
A) {3, 0}
65)
B) {9}
C) {3}
D) ∅
66) log2(3x - 2) - log2(x - 5) = 4
A)
3
13
66)
B) {6}
C)
8
38
5
D) {18}
Answer Key
Testname: AIII_SER
1) B
2) D
3) B
4) A
5) C
6) D
7) A
8) D
9) A
10) D
11) B
12) C
13) C
14) D
15) B
16) D
17) D
18) C
19) C
20) B
21) A
22) C
23) C
24) D
25) D
26) B
27) A
28) B
29) C
30) C
31) A
32) A
33) A
34) B
35) C
36) B
37) A
38) A
39) B
40) A
41) C
42) B
43) D
44) A
45) C
46) C
47) D
48) B
49) C
9
Answer Key
Testname: AIII_SER
50) B
51) C
52) D
53) D
54) D
55) D
56) D
57) A
58) A
59) C
60) A
61) A
62) D
63) B
64) B
65) C
66) B
10