Algebra 2
Name___________________________________
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Exam #5 Polynomials (and Review)
Date________________
State the number of complex roots, the possible number of real and imaginary roots, and the
possible rational roots for each equation. Then find all roots. One root has been given.
1) x 4 + 16x 3 + 62x 2 + 80x + 33 = 0; -3
A) # of complex roots: 4
Possible # of real roots: 6, 4, 2, or 0
Possible # of imaginary roots: 4, 2, or 0
Possible rational roots: ± 1, ± 3, ± 11, ± 33
Roots: {-1 mult. 2, -11, -3}
B) # of complex roots: 3
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 6, 4, 2, or 0
Possible rational roots: ± 1, ± 2, ±
1
2
Roots: {-1 mult. 2, -11, -3}
C) # of complex roots: 4
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4, 2, or 0
Possible rational roots: ± 1, ± 3, ± 11, ± 33
Roots: {-1 mult. 2, -11, -3}
D) # of complex roots: 4
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4
Possible rational roots: 0, ± 1, ± 2
1
Roots: - mult. 2, -11, -3
2
{
2) x 4 - 8x 3 + 14x 2 + 8x - 15 = 0; 5
A) # of complex roots: 2
Possible # of real roots: 6, 4, 2, or 0
Possible # of imaginary roots: 5, 3, or 1
Possible rational roots: ± 1, ± 3, ± 5, ± 15
Roots: {3, -1, 1, 5}
B) # of complex roots: 4
Possible # of real roots: 2 or 0
Possible # of imaginary roots: 4, 2, or 0
Possible rational roots: ± 1, ± 3, ± 5, ± 15
Roots: {-1 mult. 2, 1, 5}
C) # of complex roots: 2
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4
Possible rational roots: ± 1, ± 3, ± 5, ± 15
Roots: {3, -1, 1, 5}
D) # of complex roots: 4
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4, 2, or 0
Possible rational roots: ± 1, ± 3, ± 5, ± 15
Roots: {3, -1, 1, 5}
}
Simplify each expression.
3) (-8x 3 - 12) + (6x 3 + 5 - 11x 3 y 4 ) - (13 - 11x 3 )
A)
B)
C)
D)
-11x 3 y 4 + 9x 3 - 32
-6x 3 y 4 + 9x 3 - 32
-11x 3 y 4 + 9x 3 - 33
-11x 3 y 4 + 9x 3 - 20
4) (8 + 8x 2 - 3x) + (3x 2 - x - 7)
A) None of these
B) 11x 2 - 4x + 1
C) 3x 2 - 4x + 1
D) 3x 2 - 7x - 2
State the actual number of real zeros of each function.
5) f ( x) = x 4 - 2x 2 + x - 1
A) # Real Zeros: 2
B) # Real Zeros: 3
C) # Real Zeros: 1
D) None of these
6) f ( x) = x 3 - 4x 2 + 7
A) # Real Zeros: 2
B) None of these
C) # Real Zeros: 0
D) # Real Zeros: 1
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Worksheet by Kuta Software LLC
State the number of complex roots, the possible number of real and imaginary roots, and the
possible rational roots for each equation. Then find all roots.
7) x 3 + x 2 + 3x + 3 = 0
A) # of complex roots: 5
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 4, 2, or 0
1
2
Possible rational roots: 0, ± 1, ± 11, ± , ±
Roots: {-1, i 3, -i 3 }
B) None of these
C) # of complex roots: 3
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible rational roots:
0, ± 1, ± 2, ± 3, ± 4, ± 6, ± 12
Roots: {-1, i 2, -i 2 }
D) # of complex roots: 4
Possible # of real roots: 4
Possible # of imaginary roots: 2 or 0
Possible rational roots: ± 1, ± 3
Roots: {-1, i 3, -i 3 }
11
2
8) x 4 - 2x 2 - 3 = 0
A) # of complex roots: 3
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 2
Possible rational roots: ± 1, ± 3
Roots: { 3, - 3, i, -i}
B) # of complex roots: 4
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4, 2, or 0
Possible rational roots: ± 1, ± 3
Roots: { 3, - 3, i, -i}
C) # of complex roots: 3
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4, 2, or 0
Possible rational roots: 0, ± 1, ± 5
Roots: { 3, - 3, i, -i}
D) None of these
Find the value of x that makes lines u and v parallel.
9)
u
13x + 6
v
14x - 2
A) 4
C) 8
B) -12
D) 7
Find the measurement indicated in each parallelogram.
10) Find TU
11) Find m L
S
L
K
T
2x + 115
x+9
x + 115
2x + 3
M
J
V
A) 17
C) 16
A) 41°
C) 40°
U
B) 15
D) 13
B) 60°
D) 115°
Worksheet by Kuta Software LLC
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12) Find m G
H
E
9x
13) FH = x + 17
HD = 2x + 17
Find FD
G
D
H
F
A) 54°
C) 107°
F
E
22x - 6
G
A) 15
C) 8
B) 150°
D) 60°
B) 34
D) 7
Find each product.
14) (-5x + 3 y)(-3x - 5y)
A)
B)
C)
D)
15x 2 + 16xy - 15y 2
16x 2 - 24xy - 40y 2
6x 2 - 22xy + 12 y 2
16x 2 - 40y 2
16) (-5x 2 - 5x - 4)(-6x 2 - x - 3)
A)
B)
C)
D)
15) (2x - 5)(2x + 5)
4
3
A)
B)
C)
D)
4x 2 - 20x + 25
4x 2 - 25
25x 2 - 25
None of these
17) (5x - 3)(4x - 3)
2
A)
B)
C)
D)
30x + 35x + 44x + 19x + 12
-16x 4 + 28x 3 + 58x 2 - 26x - 28
-16x 4 - 20x 3 + 70x 2 - 2x - 28
-16x 4 - 6x 2 - 28
20x 2 + 9
20x 2 - 3x - 9
20x 2 - 27x + 9
20x 2 + 3x - 9
Find the area of each.
18)
19)
2 km
9 km
4.8 km
4 km
4 km
4 km
A) 16 km²
C) 4.8 km²
2 km
A) 13.8 km²
C) 8 km²
B) 6.9 km²
D) 16 km²
B) 9.6 km²
D) 19.2 km²
Describe the end behavior of each function.
20) f ( x) = -x 4 + 2x 2 - x + 2
A) None of these
B) f ( x) → -∞ as x → -∞
f ( x) → +∞ as x → +∞
C) f ( x) → -∞ as x → -∞
f ( x) → -∞ as x → +∞
D) f ( x) → +∞ as x → -∞
f ( x) → -∞ as x → +∞
21) f ( x) = -x 5 + 4x 3 - x - 2
A) Falls to the left. Rises to the right
B) Rises to the left. Rises to the right
C) Falls to the left. Falls to the right
D) Rises to the left. Falls to the right
Worksheet by Kuta Software LLC
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22) f ( x) = x 2 + 4x + 2
A) f ( x) → -∞ as x → -∞
f ( x) → +∞ as x → +∞
B) None of these
C) f ( x) → -∞ as x → -∞
f ( x) → -∞ as x → +∞
D) f ( x) → +∞ as x → -∞
f ( x) → +∞ as x → +∞
23) f ( x) = -x 4 - x 3 + 3x 2
A) Falls to the left. Falls to the right
B) None of these
C) Rises to the left. Falls to the right
D) Rises to the left. Rises to the right
Find the approximate y-values only of the local maximum and/or local minimum to the nearest
tenth.
24) f ( x) = -x 4 + 2x 2 + 2x - 3
A) Minima: None
Maxima: (1.2, 2.2)
B) Minima: (-1.9, -2.1)
Maxima: None
C) Minima: None
Maxima: (1.2, 0.2)
D) None of these
25) f ( x) = x 3 - x 2
A) Minima: (0, 0)
Maxima: (1.3, 1.2)
B) Minima: (0.7, -0.1)
Maxima: (0, 0)
C) Minima: (2.7, -5.5)
Maxima: (0, 4)
D) Minima: (0, 2)
Maxima: (2, 6)
Name each polynomial by degree and number of terms.
26) 7x 5 + 10x 4 - 5x 3 + 2x
A) fifth degree polynomial with four terms
B) None of these
C) fifth degree trinomial
D) fifth degree binomial
27) -3x 2 - 4x 3 + 6x - 1
A) quartic trinomial
B) cubic monomial
C) linear polynomial with four terms
D) cubic polynomial with four terms
State if the given binomial is a factor of the given polynomial.
28) ( x 3 + 10x 2 + 2x - 66) ¸ ( x + 9)
A) Yes
B) No
29) (4x 2 + 22x + 30) ¸ ( x + 3)
A) No
B) Yes
Find the missing side of each triangle. Round your answers to the nearest tenth if necessary.
30)
31)
13
4
3
x
12
x
A) 6
C) 5.8
A) 5
C) 17.7
B) 5
D) 2.6
B) 14
D) 11
Worksheet by Kuta Software LLC
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Divide.
32) (3x 3 - 24) ¸ (3x - 6)
33) ( x 4 - 2x 3 - 42x 2 - 50x + 16) ¸ ( x - 8)
4
3( x - 2)
1
B) x 2 + 2x + 6 +
3( x - 2)
5
C) x 2 + 2x + 6 +
3( x - 2)
2
D) x + 2x + 4
A) x 2 + 2x + 5 -
34) ( x 4 + 16x 3 + 67x 2 + 32x + 64) ¸ ( x + 8)
A) x 3 + 8x 2 + 3x + 9 +
A) x 3 + 6x 2 + 6x - 4 +
B) x 3 + 6x 2 + 6x - 2
3
x-8
3
2
D) x + 6x + 6x + 1
C) x 3 + 6x 2 + 6x +
35) (24x 2 - 7x - 6) ¸ (3x - 2)
4
x+8
3
3x - 2
2
B) 10x + 3 3x - 2
C) 8x + 3
1
D) 10x + 1 3x - 2
A) 10x -
B) x 3 + 8x 2 + 3x + 7
C) x 3 + 8x 2 + 2x + 5 +
1
x+8
D) x 3 + 8x 2 + 3x + 8
36) (4x 4 + 6x 3 - 28x 2 + 6) ¸ (4x - 2)
37) ( x 4 - 6x 3 - 13x 2 + 49x - 44) ¸ ( x - 7)
2
2x - 1
1
B) x 3 + 2x 2 - 6x - 4 +
2(2x - 1)
3
2
C) x + 2x - 6x - 3
1
D) x 3 + 2x 2 - 6x - 1 2x - 1
5
x-7
4
B) x 3 + x 2 - 6x + 9 +
x-7
1
C) x 3 + x 2 - 6x + 10 +
x-7
7
D) x 3 + x 2 - 6x + 10 +
x-7
A) x 3 + 2x 2 - 6x - 6 +
38) ( x 3 - 6x 2 - 7x) ¸ ( x + 1)
A) x 3 + x 2 - 6x + 7 +
39) (18x 3 + 24x 2 + 33x + 27) ¸ (3x + 3)
A) x 2 - 7x
B) None of these
C) x 2 - 7x - 1
D) x 2 - 7x - 2 -
1
x-8
2
3( x + 1)
1
B) 6x 2 + 2x + 10 +
x+1
5
C) 6x 2 + 2x + 10 +
3( x + 1)
D) None of these
A) 6x 2 + 2x + 7 -
4
x+1
40) (2x 4 + 12x 3 - 81x 2 - 16x - 66) ¸ ( x + 10)
41) ( x 5 + 8x 4 - 22x 3 - x 2 + 7x + 6) ¸ ( x - 2)
3
x + 10
9
B) 2x 3 - 8x 2 - x - 7 x + 10
C) None of these
6
D) 2x 3 - 8x 2 - x - 6 x + 10
1
x-2
2
B) x 4 + 10x 3 - 2x 2 - 7x - 3 +
x-2
3
C) x 4 + 10x 3 - 2x 2 - 7x - 6 x-2
4
3
2
D) x + 10x - 2x - 5x - 3
Worksheet by Kuta Software LLC
A) 2x 3 - 8x 2 - x - 7 -
A) x 4 + 10x 3 - 2x 2 - 7x - 1 +
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A polynomial function with rational coefficients has the following zeros. Find all additional
zeros.
42) -4, -1 + 2i
43) -3, -4, -1 +
A) -1 - 2i
C) None of these
B) 1 + 2i
D) 1 - 2i
6
A) None of these
C) 1 - 6
B) -1 - 6
D) 1 + 6
Expand completely.
44) ( x + 2)
A)
B)
C)
D)
4
45) ( x - y)
16 + 8x 3 + 24x 2 + 32x
None of these
x 4 + 8x 3 + 24x 2 + 32x + 16
6x 4 + 30x 3 + 80x 2 + 120x + 96
A)
B)
C)
D)
3
4x 3 - 6x 2 y + 4xy 2 - y 3
x 3 - 8x 2 y + 3xy 2 - y 3
x 3 - x 2 y + xy 2 - y 3
x 3 - 3x 2 y + 3xy 2 - y 3
Evaluate each function at the given value.
46) f ( x) = x 4 + 6x 3 + 15x 2 + 16x - 1 at x = -3
A) 5
C) None of these
47) f ( x) = x 4 - 22x 2 - 20x + 26 at x = 5
B) 6
D) 2
A) -12
C) 9
B) 7
D) 1
Find all rational zeros.
48) y = x 3 + 8
49) y = x 5 + 5x 4 + 6x 3 + 30x 2 - 27x - 135
A) {-1}
C) None of these
B) {-3}
D) {-2}
5
3
{
C) -3}
{ }
A) -
B) {-6}
D) {-5}
Write a polynomial function of least degree that has real coefficients, the following zeros, and a
leading coefficient of 1.
50) -2 mult. 2, -3i
A)
B)
C)
D)
51) -5 mult. 2,
f ( x) = x + 4x + 8x + 36x + 36
f ( x) = x 4 + 4x 3 + 10x 2 + 36x + 36
f ( x) = x 4 + 7x 3 + 13x 2 + 36x + 36
None of these
4
3
2
A)
B)
C)
D)
5
f ( x) = x 4 + 10x 3 + 20x 2 - 50x - 125
f ( x) = x 4 + 10x 3 + 18x 2 - 50x - 125
None of these
f ( x) = x 4 + 10x 3 + 20x 2 - 50x - 124
Determine if the two triangles are congruent. If they are, state how you know.
52)
A)
B)
C)
D)
ASA
Not enough information
SSS
SAS
Worksheet by Kuta Software LLC
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Find all zeros.
53) f ( x) = x 2 + 2x + 1
A)
B)
C)
D)
54) f ( x) = x 4 + 10x 2 + 21
{0 mult. 2}
{-2 mult. 2}
{-1 mult. 2}
A) {i, -i, i 7, -i 7 }
B) None of these
C) {i 3, -i 3, i 7, -i 7 }
i 6 i 6
D)
,, i 7, -i 7
2
2
None of these
{
}
State the maximum number of turns the graph of each function could make.
55) f ( x) = x 5 - 3x 3 + x + 4
A) Max # Turns: 3
B) Max # Turns: 0
C) Max # Turns: 4
D) None of these
56) f ( x) = x 3 - 4 - 3x
A) Max # Turns: 4
B) None of these
C) Max # Turns: 0
D) Max # Turns: 5
Find the measure of the angle indicated in
bold.
Solve for x. The triangles in each pair are
similar.
57)
58)
ABC ~ AKL
K
55 x + 2
10
56x
83 °
A
A) 138°
C) 112°
L
11
B) 66°
D) 124°
A
83 °
77
16x + 6
C
B
A) 6
C) 4
Complete each congruence statement by
naming the corresponding angle or side.
B) 10
D) 13
Find the length indicated.
60) Find EF
YXW @ RST
59)
x + 11
W
x+8
D
R
E
T
F
13
A) 5
C) 3
X
YX @ ?
A) TR
C) RS
Y
B) 6
D) 22
S
B) T
D) ST
Worksheet by Kuta Software LLC
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