Exam 3 245 practice
Disclaimer. The actual test does not mirror this practice. This is meant as a means to help you understand the material.
Graph the function.
2x
1) f(x) =
2
x + 4x + 3
1)
Sketch the graph of the rational function.
4x
2) f(x) =
(x + 4)(x + 2)
2)
y
x
1
Identify any vertical, horizontal, or oblique asymptotes in the graph of y = f(x). State the domain of f.
3)
y
6
4
2
-6
-4
-2
2
4
6
x
-2
-4
-6
A) Vertical: x = -3; horizontal: y = 0; (-∞, -3) ∪ (-3, ∞)
B) Vertical: x = -3; horizontal: y = 0; (-∞, 0) ∪ (0, ∞)
C) Vertical: x = 0; horizontal: y = -3; (-∞, -3) ∪ (-3, ∞)
D) Vertical: x = 0; horizontal: y = -3; (-∞, 0) ∪ (0, ∞)
Use synthetic division to perform the division.
x3 - x 2 + 3
4)
x+2
4)
Give the equation of the oblique asymptote, if any.
2
5) f(x) = x + 2x - 3
x- 4
5)
Use synthetic division to perform the division.
4
3
2
6) x - 3x - 15x - 17x - 6
x- 6
6)
Give the equation of the oblique asymptote, if any.
2
7) f(x) = x - 9x + 3
x+9
7)
Use synthetic division to perform the division.
8)
x3 - 7 x2 + 9 x - 3
2
2
2
8)
x- 1
2
9)
x4 - 3x3 - 7x2 - 14x - 5
x- 5
9)
2
3)
Use the remainder theorem and synthetic division to find f(k).
10) k = 4 + i; f(x) = x3 + 10
Solve the problem. Round to the nearest tenth unless indicated otherwise.
11) The volume of a gas varies inversely as the pressure and directly as the temperature (in
degrees Kelvin). If a certain gas occupies a volume of 2.5 liters at a temperature of 380 K
and a pressure of 20 newtons per square centimeter, find the volume when the
temperature is 456 K and the pressure is 30 newtons per square centimeter.
Use the remainder theorem and synthetic division to find f(k).
12) k = -3 + 3i; f(x) = x2 + 5x + 4
Use synthetic division to perform the division.
3
13) x - 1
x- 1
10)
11)
12)
13)
Solve the problem.
14) The weight that a horizontal beam can support varies inversely as the length of the
beam. Suppose that a 5-m beam can support 730 kg. How many kilograms can a 10-m
beam support?
15) The current I in an electrical conductor varies inversely as the resistance R of the
conductor. The current is 5 amperes when the resistance is 719 ohms. What is the current
when the resistance is 420 ohms? Round to the nearest tenth.
Solve the problem. Round to the nearest tenth unless indicated otherwise.
16) The cost of stainless steel tubing varies jointly as the length and the diameter of the
tubing. If a 5 foot length with diameter 2 inches costs $48.00, how much will a 11 foot
length with diameter 5 inches cost? Round to the nearest cent.
14)
15)
16)
Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
17) 2 ; f(x) = 5x4 + 3x2 - 4
17)
5
Use the remainder theorem and synthetic division to find f(k).
18) k = -2; f(x) = x6 + 3x5 - 3x4 + 4x3 + 3x2 - 4x - 6
18)
Express f(x) in the form f(x) = (x - k)q(x) + r for the given value of k.
19) f(x) = 2x 4 - x 3 - 15x 2 + 3x; k = -3
19)
20) f(x) = 3x 3 - x 2 + 2x + 7; k = -1
20)
Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
21) 2 + i; f(x) = x 3 + 2x2 - 6x + 8
21)
3
Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
22) f(x) = x 4 - 32x2 - 144
22)
23) f(x) = x4 - 9
23)
24) f(x) = x 4 - 45x2 - 196
24)
Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of
negative real zeros for the function.
25) 6x5 - 6x4 + 7x3 - 8 = 0
25)
Find a polynomial of lowest degree with only real coefficients and having the given zeros.
26) 4 + 6 , 4 - 6 , and 3
27) 2, -8, and 3 + 4i
26)
27)
Solve the problem. Round your answer to two decimal places.
28) The area of a circle varies directly as the square of the radius of the circle. If a circle with
a radius of 5 inches has an area of 78.5 square inches, what is the area of a circle with a
radius of 20 inches?
28)
29) The distance to the horizon varies directly as the square root of the height above ground
level of the observer. If a person can see 6 miles from a height of 25 feet, how far can a
person see from a height of 49 feet?
29)
30) The weight W of an object on the Moon varies directly as the weight E on earth. A
person who weighs 158 lb on earth weighs 31.6 lb on the Moon. How much would a
125-lb person weigh on the Moon?
30)
Find a polynomial of lowest degree with only real coefficients and having the given zeros.
31) 1 - 3, 1 + 3, and 1 + i
31)
Find the zeros of the polynomial function and state the multiplicity of each.
32) f(x) = (x2 + 14x + 45)2
32)
Find all rational zeros and factor f(x).
33) f(x) = 8x3 + 10x2 - 11x + 2
33)
34) f(x) = 10x3 + 63x2 + 17x - 6
34)
35) f(x) = 12x3 + 61x2 + 4x - 5
35)
Use the remainder theorem and synthetic division to find f(k).
36) k = -3; f(x) = 4x3- 6x2 - 4x + 22
4
36)
Find the zeros of the polynomial function and state the multiplicity of each.
37) 5x(x - 7)3 (x2 - 16)
37)
Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
38) -5 - 4i; f(x) = x2 + 10x + 41
38)
Use the remainder theorem and synthetic division to find f(k).
39) k = -3; f(x) = 6x4+ 10x3 + 2x2 - 6x + 35
39)
Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
40) -2; f(x) = -6x3 + 3x2 + x - 58
40)
Use the remainder theorem and synthetic division to find f(k).
41) k = 2 + i; f(x) = x3 + 10
41)
Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
42) 8x3 + 36x2 - 19x + 5; x + 5
42)
Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
43) 7i; f(x) = x3 + 2x2 + 49x + 98
43)
For the function as defined that is one-to-one, graph f and f -1 on the same axes.
44) f(x) =
x+6
44)
y
10
5
-10
-5
5
10 x
-5
-10
Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
45) 1 ; f(x) = 2x4 - 21x3 + 3x + 1
45)
2
Factor f(x) into linear factors given that k is a zero of f(x).
46) f(x) = x3 - (4 + 2i)x2 + (-5 + 8i)x + 10i; k = 2i
46)
5
For the function as defined that is one-to-one, graph f and f -1 on the same axes.
47) f(x) = 5 x + 4
3
47)
y
10
5
-10
-5
10 x
5
-5
-10
Factor f(x) into linear factors given that k is a zero of f(x).
48) f(x) = 6x3 + 37x2 + 32x - 15; k = 1
3
48)
Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
49) -3x3 + 4x2 - 3x + 2; x + 2
49)
The graph of a function f is given. Use the graph to find the indicated value.
50) f-1(3)
50)
y
8
6
4
2
2
4
6
8
x
Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
51) 3i; f(x) = x3 + 4x2 + 9x + 36
51)
6
For the function as defined that is one-to-one, graph f and f -1 on the same axes.
52) f(x) = 7
x
52)
y
10
5
-10
-5
5
10 x
-5
-10
For the polynomial, one zero is given. Find all others.
53) P(x) = x 3 - 3x2 - 5x + 39; -3
53)
Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
54) 3x2 - 3x + 18; x - 2
54)
Find the domain and range of the inverse of the given function.
55) f(x) = x - 9
55)
Give all possible rational zeros for the following polynomial.
56) P(x) = 2x 3 - 5x 2 + 7x - 23
56)
Factor f(x) into linear factors given that k is a zero of f(x).
57) f(x) = x3 - 2x2 - 36x + 72 ; k = 6
57)
Give all possible rational zeros for the following polynomial.
58) P(x) = x 3 - 9x2 + 7x - 24
58)
Find all rational zeros and factor f(x).
59) f(x) = 4x3 - 28x2 - x + 7
59)
Factor f(x) into linear factors given that k is a zero of f(x).
60) f(x) = x 3 - 48x - 128; k = -4 (multiplicity 2)
60)
7
For the polynomial, one zero is given. Find all others.
61) P(x) = x 3 - 2x2 - 11x + 52; -4
61)
Find the domain and range of the inverse of the given function.
62) f(x) = 7 - x2 ; x ≥ 0
62)
Give all possible rational zeros for the following polynomial.
63) P(x) = 2x 3 + 6x2 + 5x - 8
63)
Find the zeros of the polynomial function and state the multiplicity of each.
64) 5x(x - 7)3 (x2 - 4)
64)
Find all rational zeros and factor f(x).
65) f(x) = x3 - 8x2 + 9x + 18
65)
Find the zeros of the polynomial function and state the multiplicity of each.
66) f(x) = 3(x + 6)2(x - 6)3
66)
Find a polynomial of lowest degree with only real coefficients and having the given zeros.
67) 3 + 3 , 3 - 3 , and 3
67)
68) -
7,
7, and -3i
68)
Find the future value.
69) $5481 invested for 4 years at 4% compounded annually
69)
Solve the equation.
x +4
70) ex - 1 = 1
e5
70)
71) m-4 = 1
81
71)
Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of
negative real zeros for the function.
72) -6x4 - 8x3 - 7x2 - 5x + 7 = 0
72)
73) -9x4 + 3x3 - 7x2 + 7x - 9 = 0
73)
8
Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
74) f(x) = x4 - 36
74)
Find the zeros of the polynomial function and state the multiplicity of each.
75) 16x7 + x 5
75)
Find the equation that the given graph represents.
76)
76)
A) P(x) = 2x3 + x 2 - 3x + 4
C) P(x) = 2x4 - x 2 + 4
B) P(x) = -x3 - x - 4
D) P(x) = -x4 + x 2 - 3x - 4
Find the future value.
77) $3443.61 invested for 11 years at 4% compounded monthly
Solve the problem.
78) Find the required annual interest rate, to the nearest tenth of a percent, for $1113 to grow
to $1830 if interest is compounded quarterly for 5 years.
Sketch the graph of the polynomial function. Label at least two points on the graph.
79) f(x) = x 4 + 3
y
10
5
-10
-5
5
10 x
-5
-10
9
77)
78)
79)
80) f(x) = -(x + 2)4
80)
y
10
5
-10
-5
5
10 x
-5
-10
Find the equation that the given graph represents.
81)
81)
A) P(x) = -x6 + 20x 4 - 100x 2 + 100
C) P(x) = x5 - 10x 4 - 100x 2 + 100
B) P(x) = -x6 + 10x 4 - 100x 2 - 100
D) P(x) = -x5 - 20x 4 - 100x 2 + 100x
Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
82) f(x) = x 4 - 21x2 - 100
82)
Find a polynomial of lowest degree with only real coefficients and having the given zeros.
83) 8, -14, and 3 + 8i
Solve the problem.
84) Find the required annual interest rate, to the nearest tenth of a percent, for $168 to grow
to $547 if interest is compounded weekly for 10 years. Assume exactly 52 weeks per
year.
10
83)
84)
Answer the question.
85) How many positive real zeros does this graph have?
85)
Solve the problem.
86) If x varies inversely as y 2, and x = 6 when y = 8, find x when y = 4.
86)
87) If f varies jointly as q 2 and h, and f = 144 when q = 4 and h = 3, find q when f = 72 and h =
6.
87)
Determine whether or not the function is one-to-one. Give reason for answer.
88)
88)
y
10
10 x
-10
-10
Find the horizontal asymptote of the given function.
2
89) h(x) = 16x
8x2 - 3
89)
Solve the problem.
90) If m varies directly as x and y, and m = 24 when x = 6 and y = 9, find m when x = 15 and
y = 8.
90)
Give the domain and range for the rational function. Use interval notation.
1
91) f(x) =
+1
(x - 2)2
91)
11
Use the graph to answer the question.
92) Find the horizontal and vertical asymptotes of the rational function graphed below.
92)
y
6
4
2
-6
-4
-2
2
4
6
x
-2
-4
-6
Solve the problem.
93) A(x) = -0.015x 3 + 1.05x gives the alcohol level in an average person's blood x hrs after
drinking 8 oz of 100-proof whiskey. If the level exceeds 1.5 units, a person is legally
drunk. Would a person be drunk after 2 hours?
Use the boundedness theorem to determine whether the statement is true or false.
94) The polynomial f(x) = x5 + 10x4 - 25x3 - 10x2 + 24x has no real zero greater than 3.
93)
94)
Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the
numbers given.
95) f(x) = 10x4 - 8x3 + 6x - 7; -1 and 0
95)
Find the correct end behavior diagram for the given polynomial function.
96) P(x) = -2x6 + 3x5 - x 2 - 9x + 2
96)
Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
97) Zeros of -3, 2, 4 and P(1) = 12
97)
98) Zeros of -2, 1, 0 and P(2) = 40
98)
Find the correct end behavior diagram for the given polynomial function.
99) P(x) = 3x7 + 2x2 - 8
99)
Find a polynomial of lowest degree with only real coefficients and having the given zeros.
100) 6 + 2i and 6 - 2i
100)
Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of
negative real zeros for the function.
101) 7x3 - 5x2 + 3x + 4 = 0
101)
12
102) 8x5 - 9x4 + 6x3 - 6 = 0
102)
Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
103) f(x) = x 3 - 8x2 + 17x - 30
103)
Sketch the graph of the polynomial function.
104) f(x) = x 3 - 2 Label at least two points on the graph.
104)
y
10
5
-10
-5
5
10 x
-5
-10
105) f(x) = x 4 + 2
Label at least two point on the graph.
105)
y
10
5
-10
-5
5
10 x
-5
-10
Find the correct end behavior diagram for the given polynomial function.
106) P(x) = 7x3 + 8x2 - 4x + 2
13
106)
Graph the polynomial function. Factor first if the expression is not in factored form.
107) f(x) = 2x(x + 2)(x - 1) Label at least two points on the graph.
107)
y
10
5
-5
5
x
-5
-10
Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the
numbers given.
108) f(x) = 6x3 + 6x2 - 6x + 5; -2 and -1
108)
Use the boundedness theorem to determine whether the statement is true or false.
109) The polynomial f(x) = x4 - 9x3 - 22x2 has no real zero greater than 8.
109)
Give the domain and range for the rational function. Use interval notation.
110) f(x) = 2 + 3
x
110)
Find any vertical asymptotes.
111) h(x) = (x - 5)(x + 2)
x2 - 1
111)
Find the horizontal asymptote of the given function.
2
112) f(x) = 3x + 2
3x2 - 2
112)
14
Sketch the graph of the rational function.
113) f(x) = x - 4 Label at least two points on the graph and draw any asymptotes.
x +5
Solve the problem. Round your answer to two decimal places.
114) The period of vibration P for a pendulum varies directly as the square root of the length
L. If the period of vibration is 4.5 sec when the length is 81 inches, what is the period
when L = 5.0625 inches?
Solve the problem.
115) The weight of a body above the surface of the earth is inversely proportional to the
square of its distance from the center of the earth. What is the effect on the weight when
the distance is multiplied by 3?
Determine whether or not the function is one-to-one. Give a reason for your answer.
116)
113)
114)
115)
116)
y
10
5
-10
-5
5
10 x
-5
-10
If f is one-to-one, find an equation for its inverse.
117) f(x) = -4x + 1
117)
15
118) f(x) =
3
x- 6
118)
Determine whether or not the function is one-to-one.
119) f(x) = 2x3 - 7
120) f(x) =
119)
25 - x 2
120)
If f is one-to-one, find an equation for its inverse.
121) f(x) = 9
x+8
121)
122) f(x) = 6x3 - 7
122)
Decide whether or not the functions are inverses of each other.
123) f(x) = 6x2 + 2, domain [0, ∞); g(x) =
124) f(x) =
x-2
, domain [2, ∞)
6
1 , g(x) = 7x + 1
x+7
x
123)
124)
125)
125)
y
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
x
16
Answer Key
Testname: 245 3_2T4_1EXAMP
1)
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2
4
6
8 10
4
6
8
2)
20
16
12
8
4
y
-8 -6 -4 -2
-4
-8
-12
-16
-20
2
x
3) A
4) x2 - 3x + 6 + - 9
x+2
5) y = x + 6
6) x3 + 3x2 + 3x + 1
7) y = x - 18
8) x2 - 3x + 3
9) x3 + 2x2 + 3x + 1
10) 62 + 47i
11) 2.0 liters
12) -11 - 3i
13) x2 + x + 1
14) 365 kg
15) 8.6 amperes
16) $264.00
17) No
18) -98
19) f(x) = (x + 3)(2x3 - 7x 2 + 6x - 15) + 45
20) f(x) = (x + 1)(3x2 - 4x + 6) + 1
21) No
22) -2i, 2i, 6, -6
23) 3, - 3i, 3i, 24) -2i, 2i, 7, -7
3
17
Answer Key
Testname: 245 3_2T4_1EXAMP
25) Positive (3, 1), negative (0)
26) f(x) = x3 - 11x2 + 34x - 30
27) f(x) = x 4 - 27x2 + 246x - 400
28) 1256 square inches
29) 8.4 miles
30) 25 lb
31) f(x) = x4 - 4x3 + 4x2 - 4
32) -5, multiplicity 2; -9, multiplicity 2
33) 1 , 1 , -2; f(x) = (2x - 1)(4x - 1)(x + 2)
2 4
34) - 1 , 1 , -6; f(x) = (2x + 1)(5x - 1)(x + 6)
2 5
35) - 1 , 1 , -5; f(x) = (3x + 1)(4x - 1)(x + 5)
3 4
36) -128
37) Multiplicity 1 : 0
Multiplicity 1 : ±4
Multiplicity 3 : 7
38) Yes
39) 287
40) Yes
41) 12 + 11i
42) Yes
43) Yes
44)
y
10
5
-10
-5
5
10 x
-5
-10
45) Yes
46) f(x) = (x - 2i)(x + 1)(x - 5)
18
Answer Key
Testname: 245 3_2T4_1EXAMP
47)
y
10
5
-10
-5
5
10 x
5
10 x
-5
-10
48) (3x - 1)(2x + 3)(x + 5)
49) No
50) 2
3
51) Yes
52)
y
10
5
-10
-5
-5
-10
Function is its own inverse
53) 3 + 2i, 3 - 2i
54) No
55) Domain: [0, ∞); range: [9, ∞)
56) ±1, ±23, ±1/2, ± 23/2
57) (x - 6)(x - 2)(x + 6)
58) ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24
59) 1 , - 1 , 7; f(x) = (2x - 1)(2x + 1)(x - 7)
2
2
60) f(x) = (x + 4)2(x - 8)
61) 3 + 2i, 3 - 2i
62) Domain: (-∞, 7]; range: range: [0, ∞)
63) ±1, ±1/2, ±2, ±4, ±8
64) Multiplicity 1 : 0
Multiplicity 1 : ±2
Multiplicity 3 : 7
65) 3, 6, -1; f(x) = (x - 3)(x - 6)(x + 1)
19
Answer Key
Testname: 245 3_2T4_1EXAMP
66) -6, multiplicity 2; 6, multiplicity 3
67) f(x) = x3 - 9x2 + 24x - 18
68) f(x) = x 4 + 2x2 - 63
69) $6411.99
70) - 19
6
71) -3, 3
72) Positive (1), negative (3, 1)
73) Positive (4, 2, 0), negative (0)
74) 6, - 6i, 6i, - 6
75) Multiplicity 5 : 0
Multiplicity 1 : (±1/4)i
76) A
77) $5343.01
78) 10.1%
79)
y
10
5
-10
-5
5
10 x
5
10 x
-5
-10
80)
y
10
5
-10
-5
-5
-10
81) A
82) -2i, 2i, 5, -5
83) f(x) = x 4 - 75x2 + 1110x - 8176
84) 11.8%
85) 2
86) 24
87) 2
20
Answer Key
Testname: 245 3_2T4_1EXAMP
88) Yes
89) y = 2
90) 160
3
91) Domain: (-∞, 2) ∪ (2, ∞); Range: (1, ∞)
92) Horizontal: y = -1; vertical: x = ±4
93) Yes
94) True
95) f(-1) = 5 and f(0) = -7
96)
97) P(x) = x 3 - 3x2 - 10x + 24
98) P(x) = 5x3 + 5x2 - 10x
99)
100) x2 - 12x + 40
101) Positive (2, 0), negative (1)
102) Positive (3, 1), negative (0)
103) 6, 1 + 2i, 1 - 2i
104)
y
10
5
-10
-5
5
10 x
5
10 x
-5
-10
105)
y
10
5
-10
-5
-5
-10
106)
21
Answer Key
Testname: 245 3_2T4_1EXAMP
107)
y
10
5
-5
5
x
-5
-10
108) f(-2) = -7 and f(-1) = 11
109) False
110) Domain: (-∞, 0) ∪ (0, ∞); Range: (-∞, 3) ∪ (3, ∞)
111) x = 1, x = -1
112) y = 1
113)
8
6
4
2
-8 -6 -4 -2
-2
-4
-6
-8
2
4
6
8 10
114) 1.125 sec
115) The weight is divided by 9.
116) No
117) f-1(x) = - 1 x + 1
4
4
118) f-1(x) = x3 + 6
119) Yes
120) No
121) f-1(x) = -8x + 9
x
3 x+7
122) f-1(x) =
6
123) Yes
124) No
125) No
22