Math 112 Chapter 3 Practice Test
Name___________________________________
Classify the polynomial as constant, linear, quadratic,
cubic, or quartic, and determine the leading term, the
leading coefficient, and the degree of the polynomial.
B)
1) f(x) = 13x4 - 9 + 0.18x2 - 8x
2) f(x) = -13 - x2
Find the correct end behavior diagram for the given
polynomial function.
2
3) P(x) = x7 + 7x2 - 6
3
-6
4) P(x) = 3x6 - x5 + 8x2 - 1
5) P(x) = -3x4 - x3 + x2 - 4x + 3
C)
Use the leading-term test to match the function with the
correct graph.
6) f(x) = -0.7x6 - x5 + 6x4 - 5x3 - 5x2 + x - 3
A)
-6
-6
D)
-6
1
7) f(x) = 4x + 3x2 - 7x3
D)
A)
-5
-5
Use substitution to determine whether the given number is
a zero of the given polynomial.
B)
8) -2; P(x) = -2x3 + x2 + 2x + 3
9) -1; P(x) = x4 - 7x2 - 7
10) 3; P(x) = -x4 - 3x2 + 8x + 84
-5
Find the zeros of the polynomial function and state the
multiplicity of each.
11) f(x) = 5(x + 9)2 (x - 9)3
12) f(x) = 2x(x - 4)(x + 11)(2x - 1)
C)
13) f(x) = x4 - 25x2 + 144
14) f(x) = x3 + x2 - 2x - 2
-5
2
Graph the function.
18) f(x) = x3 + 2x2 - x - 2
15) P(x) = (-2x - 2)(x - 1)2
10
y
10
8
6
y
4
2
5
-10 -8 -6 -4 -2-2
-10
-5
5
2
4 6
8 10
x
-4
-6
-8
x
-10
-5
Evaluate the function for the given values of a and b. Then
use the intermediate value theorem to determine which of
the statements below is true.
-10
16) f(x) = x3 - 4x2
19) a = 2 and b = 3
10
P(x) = 2x5 - 9x3 - 2x2 + 1
y
20) a = 1 and b = 2
5
-10
P(x) = 5x3 + 2x - 10
-5
5
21) a = 0, b = 4
f(x) = x3 + 2x2 + 3x + 2
x
-5
Use long division to determine whether the binomial is a
factor of f(x).
22) f(x) = x3 + 11x2 + 20x - 32; x - 2
-10
23) f(x) = x3 - 9x2 + 8x + 64; x + 4
17) f(x) = -x4 - 2x2
24) f(x) = 4x3 - 21x2 - 9x + 70; x + 4
y
20
15
25) f(x) = x4 - x3 - 3x2 + 4x + 7; x + 2
10
5
-10
-5
5
A polynomial P(x) and a divisor d(x) are given. Use long
division to find the quotient Q(x) and the remainder R(x)
when P(x) is divided by d(x), and express P(x) in the form
d(x)· Q(x) + R(x).
10 x
-5
-10
26) P(x) = x3 - x2 + 5
d(x) = x + 2
-15
-20
27) P(x) = 2x 4 - x3 - 15x2 + 3x
d(x) = x + 3
28) P(x) = x3 - 3
d(x) = x + 3
3
Use synthetic division to find the quotient and the
remainder.
Graph the polynomial function. Use synthetic division and
the remainder theorem to find the zeros.
29) (2x 3 + 3x 2 + 4x - 10) ÷ (x + 1)
45) f(x) = x3 + 2x2 - 15x - 36
60
y
30) (3x 4 - 9x 3 + 2x 2 - 6x) ÷ (x - 3)
31) (2x 5 - x4 + 3x 2 - x + 5) ÷ (x - 1)
6 x
-6
32) (x3 - 3) ÷ (x - 1)
33) (x5 - 2x4 - 17x3 + 12x2 - 13x + 17) ÷ (x - 5)
-60
Use synthetic division to find the function value.
34) f(x) = 2x3 + 5x2 - 5x + 13; find f(3)
46) f(x) = x3 + 9x2 - 108
120
35) f(x) = x4 + 16; find f(3).
y
36) f(x) = x2 - 3x - 3; find f(-4 + 2i)
Using synthetic division, determine whether the numbers
are zeros of the polynomial.
10 x
-10
37) 5, 3; f(x) = x3 - 9x2 + 26x - 24
-120
38) 0, -2; P(x) = -x4 + 6x2 + 2x - 4
47) f(x) = -x3 - 4x2 + 3x + 18
39) 2i, -i, -4; f(x) = x3 + 4x2 + x + 4
200
y
Factor the polynomial f(x). Then solve the equation f(x) = 0.
40) f(x) = x3 + 2x2 - 23x - 60
41) f(x) = x3 + 12x2 + 44x + 48
x
42) f(x) = x3 + 4x2 - 17x - 60
43) f(x) = x4 - 4x3 - 7x2 + 22x + 24
-200
44) f(x) = x4 + 2x3 - 19x2 - 8x + 60
Find the requested polynomial.
48) Find a polynomial function of degree 3 with 5, i,
-i as zeros.
49) Find a polynomial function of degree 3 with 1 +
3i, 1 - 3i, -1 as zeros.
4
Give all possible rational zeros for the polynomial.
50) Find a polynomial of degree 4 having the
following zeros:
-9 (multiplicity 2), 10, - 10
64) P(x) = 2x 3 + 7x2 + 11x - 8
65) P(x) = -2x4 + 2x3 + 6x2 + 18
51) Find a polynomial of degree 5 with -2 as a zero
of multiplicity 3, 0 as a zero of multiplicity 1,
and 3 as a zero of multiplicity 1.
66) P(x) = 22x3 + 88x2 + 2x - 11
Given the polynomial function f(x), find the rational zeros,
then the other zeros (that is, solve the equation f(x) = 0), and
factor f(x) into linear factors.
Provide the requested response.
52) Suppose that a polynomial function of degree 4
with rational coefficients has 6,
4, 3i as zeros. Find the other zero.
67) f(x) = x3 - 27x - 54
68) f(x) = x4 + 15x3 + 49x2 - 15x - 50
53) Suppose that a polynomial function of degree 4
with rational coefficients has -4 + 3i, 3 - 5 as
zeros. Find the other zeros.
69) f(x) = x3 - 9x2 + 12x + 14
Find only the rational zeros.
54) Suppose that a polynomial function of degree 5
with rational coefficients has -3, 5, -5, 3 - i as
zeros. Find the other zero(s).
70) P(x) = x3 - 5x2 - 4x + 20
71) x4 + 36
55) Suppose that a polynomial function of degree 5
with rational coefficients has 6,
-3 + 4i, 4 - 2 as zeros. Find the other zeros.
72) x4 - 8x3 + 4x2 + 24x - 21
73) x4 + 4x3 + 15x2 - 16x - 44
Find a polynomial function of lowest degree with rational
coefficients that has the given numbers as some of its zeros.
56) -8i, 7
Use Descartesʹ Rule of Signs to determine the possible
number of positive real zeros and the possible number of
negative real zeros for the function.
57) 2 - i, -6
74) P(x) = 3x8 + 9x6 + 7x4 + 9x2 + 5
58) 1 - 3 , 1 + i
75) P(x) = 8x5 - 8x4 + 5x3 - 7
59) 2, -8, 3 + 4i.
76) P(x) = 5x6 - 9x4 - 7x3 + 9x2 - 8x
Given that the polynomial function has the given zero, find
the other zeros.
60) f(x) = x3 - 3x2 - 5x + 39; -3
61) f(x) = x4 - 21x2 - 100; -2i
62) f(x) = x3 - 64; 4
63) f(x) = x4 - 36; 6
5
Sketch the graph of the polynomial function. Use the
rational zeros theorem when finding the zeros.
C)
y
77) f(x) = 2x3 - 3x2 - 14x + 21
8
y
30
20
-8
8
x
8
x
10
-8
-5
-4
-3
-2
-1
1
2
3
4
5 x
D)
-10
y
-20
8
-30
-8
Match the equation with the appropriate graph.
6
78) f(x) = 2
x + 1
-8
A)
y
79) f(x) = 8
3x2
x2 - 9
A)
10
-8
8
y
x
-8
10 x
-10
B)
y
-10
8
B)
10
-8
8
y
x
-8
10 x
-10
-10
6
Find the oblique asymptote, if any, of the rational function.
x + 7
87) f(x) = 2x2 + 7x - 4
C)
10
y
88) f(x) = x2 + 4x + 4
x + 9
89) f(x) = 8x3 - 5
x2 + 6
10 x
-10
-10
State the domain of the rational function.
8
90) f(x) = 6 - x
D)
10
y
91) f(x) = x - 4
x2 + 2
10 x
-10
92) f(x) = x - 7
x2 - 36
-10
Graph the function, showing all asymptotes (those that do
not correspond to an axis) as dashed lines. List the x - and
y-intercepts.
2x - 1
93) f(x) = x
Find the vertical asymptote(s) of the graph of the given
function.
x + 5
80) g(x) = x - 3
81) f(x) = x - 9
x2 + 8
82) f(x) = 83) f(x) = x - 11
x2 - 4
x2 + 2x - 8
x2 - 3x - 10
94) f(x) = Find the horizontal asymptote, if any, of the rational
function.
(x - 1)(x + 1)
84) f(x) = x2 - 1
85) f(x) = 86) f(x) = x2 + 9x + 4
8 - x2
x2 + 9x - 6
x - 6
7
1
x - 2
95) f(x) = 4x - 1
x
Solve the given inequality (a related function is graphed).
98) -x2 + 2x ≤ -6x + 12
y
10
-10
10
x
10
x
-10
96) f(x) = x2 - 16
x - 4
x-intercepts: (2, 0), (6, 0)
99)
7
≤ 0
2
x - 16
y
10
-10
97) f(x) = x + 3
-10
x - 3 3
critical values: -4, 4
Solve.
100) x2 - 3x - 28 < 0
101) 10 - x2 ≥ 0
102) x2 + 6x + 9 ≤ 0
103) x3 + 5x2 - 9x - 45 ≥ 0
104) x3 - 2x ≤ 24 - 5x2
8
105)
1
≤ 0
x - 2
106)
x + 11
< 7
x + 9
107)
x - 6 x + 4
- ≤ 0
x + 5 x - 3
108)
x + 4 3x + 2
> x - 3 2x + 1
109)
3
4
≤ 2
2
x - 9 x + 8x + 15
110)
5
4
< 2
2
x - 1 x - 9
9
Answer Key
Testname: MATH 112 CHAPTER 3 PRACTICE TEST
1) Quartic; 13x4 ; 13; 4
2) Quadratic; -x2 ; -1; 2
3)
4)
5)
6) C
7) D
8) No
9) No
10) Yes
11) -9, multiplicity 2; 9, multiplicity 3
1
12) -11, multiplicity 1; 0, multiplicity 1; , multiplicity 1; 4, multiplicity 1
2
13) -4, multiplicity 1; 4, multiplicity 1; -3, multiplicity 1; 3, multiplicity 1
14) -1, multiplicity 1; 2, multiplicity 1; - 2, multiplicity 1
15)
10
y
5
-10
-5
5
x
5
x
-5
-10
16)
10
y
5
-10
-5
-5
-10
10
Answer Key
Testname: MATH 112 CHAPTER 3 PRACTICE TEST
17)
y
20
15
10
5
-10
-5
10 x
5
-5
-10
-15
-20
18)
y
10
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
2
4 6
8 10
x
-6
-8
-10
19) P(2 ) and P(3) have opposite signs, therefore the function P has a real zero between 2 and 3.
20) P(1) and P(2) have opposite signs, therefore P has a real zero between 1 and 2.
21) f(0) and f(4) have the same sign, therefore the intermediate value theorem cannot be used to determine whether f has a
real zero between 0 and 4
22) No
23) No
24) No
25) No
26) (x + 2)·(x2 - 3x + 6) - 7
27) (x + 3) · (2x3 - 7x 2 + 6x - 15) + 45
28) (x + 3)·(x2 - 3x + 9) - 30
29) Q(x) = (2x2 + x + 3); R(x) = -13
30) Q(x) = (3x3 + 2x); R(x) = 0
31) Q(x) = (2x4 + x3 + x2 + 4x + 3); R(x) = 8
32) Q(x) = x2 + x + 1; R(x) = -2
33) Q(x) = x4 + 3x3 - 2x2 + 2x - 3 ; R(x) = 2
34) 97
35) 97
36) 21 - 22i
37) No; yes
38) No; yes
39) No; yes; yes
40) (x + 3)(x + 4)(x - 5) ; -3, -4, 5
41) (x + 2)(x + 4)(x + 6) ; -2, -4, -6
11
Answer Key
Testname: MATH 112 CHAPTER 3 PRACTICE TEST
42) (x + 3)(x - 4)(x + 5) ; -3, 4, -5
43) (x + 2)(x - 3)(x + 1)(x - 4); -2, 3, -1, 4
44) (x - 2)(x + 2)(x - 3)(x + 5); 2, -2, 3, -5
45) -3 (multiplicity 2), 4;
60
y
6 x
-6
-60
46) -6 (multiplicity 2), 3;
120
y
10 x
-10
-120
47) -3 (multiplicity 2), 2;
200
y
x
-200
48) f(x) = x3 - 5x2 + x - 5
49) f(x) = x3 - x2 + 8x + 10
50) f(x) = x4 + 18x3 + 71x2 - 180x - 810
51) f(x) = x5 + 3x4 - 6x3 - 28x2 - 24x
52) - 3i
53) -4 - 3i , 3 + 5
54) 3 + i
55) -3 - 4i, 4 + 2
56) f(x) = x4 + 57x2 - 448
57) f(x) = x3 + 2x2 - 19x + 30
12
Answer Key
Testname: MATH 112 CHAPTER 3 PRACTICE TEST
58) f(x) = x4 - 4x3 + 4x2 - 4
59) f(x) = x4 - 27x2 + 246x - 400
60) 3 + 2i, 3 - 2i
61) 2i, 5, -5
62) -2 + 2 3 i, -2 - 2 3 i
63) - 6 i, 6 i, - 6
64) ±1, ±1/2, ±2, ±4, ±8
65) ±1, ±1/2, ±2, ±3, ±3/2, ±6, ±9, ±9/2, ±18
66) ±1, ±1/2, ±11, ±11/2, ±1/11, ±1/22
67) -3, multiplicity 2; 6; f(x) = (x + 3)2 (x - 6)
68) -10, -5, -1, 1; f(x) = (x + 10)(x + 5)(x + 1)(x - 1)
69) 7, 1 + 3 , 1 - 3 ; f(x) = (x - 7)(x - 1 - 3)(x - 1 + 3)
70) 2, 5, -2
71) No rational zeros
72) 7, 1
73) No rational zeros
74) 0 positive; 0 negative
75) 1 or 3 positive; 0 negative
76) 1 or 3 positive; 0 or 2 negative
30
y
20
10
-5
-4
-3
-2
-1
1
2
3
4
5 x
-10
-20
77)
-30
78) C
79) C
80) x = 3
81) None
82) x = 2, x = -2
83) x = -2, x = 5
84) y = 1
85) y = -1
86) None
87) None
88) y = x - 5
89) y = 8x
90) (-∞, 6) ∪ (6, ∞)
91) (-∞, ∞)
92) (-∞, - 6) ∪ (- 6, 6) ∪ (6, ∞)
13
Answer Key
Testname: MATH 112 CHAPTER 3 PRACTICE TEST
93) x-intercept: 1
, 0 , no y-intercepts ;
2
10
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
2 4
6 8 10
-6
-8
-10
94) No x-intercepts, y-intercept: 0, - 1
;
2
10
8
6
4
2
-10 -8 -6 -4 -2-2
2 4
6 8 10
-4
-6
-8
-10
95) x-intercept: 1
, 0 , no y-intercepts ;
4
10
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
2 4
6 8 10
-6
-8
-10
96) x-intercept: -4, 0 , y-intercept: 0, 4 ;
10
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
2 4
6 8 10
-6
-8
-10
14
Answer Key
Testname: MATH 112 CHAPTER 3 PRACTICE TEST
97) x-intercept: (-3, 0) , y-intercept: 0, - 1
;
9
10
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
2 4
6 8 10
-6
-8
-10
98) (-∞, 2] ∪ [6, ∞)
99) (-4, 4)
100) (-4, 7)
101) [- 10, 10]
102) {-3}
103) [-5, -3] ∪ [3, ∞)
104) (-∞, -4] ∪ [-3, 2]
105) (-∞, 2)
26
106) -∞, -9 ∪ - , ∞
3
107) -5, - 1
∪ (3, ∞)
9
108) 8 - 74, - 1
∪ 3, 8 + 74
2
109) (-∞, -5) ∪ (-3, 3) ∪ [27, ∞)
110) (- 41, -3) ∪ (-1, 1) ∪ (3, 41)
15