Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
eSolutionsManual-PoweredbyCogneroPage1
5-7 Roots and Zeros
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
eSolutionsManual-PoweredbyCogneroPage2
5-7 Roots and Zeros
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
eSolutionsManual-PoweredbyCogneroPage3
5-7 Roots and Zeros
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
eSolutionsManual-PoweredbyCogneroPage4
5-7 Roots and Zeros
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
eSolutionsManual-PoweredbyCogneroPage5
5-7 Roots and Zeros
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
eSolutionsManual-PoweredbyCogneroPage6
5-7 Roots and Zeros
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
eSolutionsManual-PoweredbyCogneroPage7
5-7 Roots and Zeros
Solve each equation. State the number and type
of roots.
1.x
2
– 3x – 10 = 0
ANSWER:
–2, 5; 2 real
2.x
3
+ 12x
2
+ 32x =0
ANSWER:
–8, –4, 0; 3 real
3.16x
4
– 81 = 0
ANSWER:
2 real, 2 imaginary
4.0 = x
3
– 8
ANSWER:
1 real, 2 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
5.f (x) = x
3
– 2x
2
+ 2x – 6
ANSWER:
3 or 1; 0; 0 or 2
6.f (x) = 6x
4
+ 4x
3
– x
2
– 5x – 7
ANSWER:
1; 1 or 3; 0 or 2
7.f (x) = 3x
5
– 8x
3
+ 2x – 4
ANSWER:
1 or 3; 0 or 2; 0, 2, or 4
8.f (x) = –2x
4
– 3x
3
– 2x – 5
ANSWER:
0; 0 or 2; 2 or 4
Find all zeros of each function.
9.f (x) = x
3
+ 9x
2
+ 6x – 16
ANSWER:
–8, –2,1
10.f (x) = x
3
+ 7x
2
+ 4x + 28
ANSWER:
–7, –2
i
, 2
i
11.f (x) = x
4
– 2x
3
– 8x
2
– 32x – 384
ANSWER:
–4, 6, –4
i
, 4
i
12.f (x) = x
4
– 6x
3
+ 9x
2
+ 6x – 10
ANSWER:
–1, 1, 3 –
i
, 3 +
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
13.4, –1, 6
ANSWER:
x
3
– 9x
2
+ 14x + 24
14.3, –1, 1, 2
ANSWER:
x
4
– 5x
3
+ 5x
2
+ 5x – 6
15.–2, 5, –3
i
ANSWER:
x
4
– 3x
3
– x
2
– 27x – 90
16.–4, 4 +
i
ANSWER:
x
3
– 4x
2
– 15x + 68
Solve each equation. State the number and type
of roots.
17.2x
2
+ x – 6 = 0
ANSWER:
18.4x
2
+ 1 = 0
ANSWER:
19.x
3
+ 1 = 0
ANSWER:
20.
2x
2
–
5x + 14 = 0
ANSWER:
imaginary
21.–
3x
2
–
5x + 8 = 0
ANSWER:
22.
8x
3
–
27 = 0
ANSWER:
23.
16x
4
–
625 = 0
ANSWER:
24.
x
3
–
6x
2
+ 7x = 0
ANSWER:
25.
x
5
–
8x
3
+ 16x = 0
ANSWER:
–
2,
–
2, 0, 2, 2; 5 real
26.
x
5
+ 2x
3
+ x = 0
ANSWER:
0,
–
i,
–
i, i, i; 1 real, 4 imaginary
State the possible number of positive real
zeros, negative real zeros, and imaginary zeros
of each function.
27.
f (x) = x
4
–
5x
3
+ 2x
2
+ 5x + 7
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
28.
f (x) = 2x
3
–
7x
2
–
2x + 12
ANSWER:
0 or 2; 1; 0 or 2
29.
f (x) =
–
3x
5
+ 5x
4
+ 4x
2
–
8
ANSWER:
0 or 2; 1; 2 or 4
30.
f (x) = x
4
–
2x
2
–
5x + 19
ANSWER:
0 or 2; 0 or 2; 0, 2, or 4
31.
f (x) = 4x
6
–
5x
4
–
x
2
+ 24
ANSWER:
0 or 2; 0 or 2; 2, 4, or 6
32.
f (x) =
–
x
5
+ 14x
3
+ 18x
–
36
ANSWER:
0 or 2; 1; 2 or 4
Find all zeros of each function.
33.
f (x) = x
3
+ 7x
2
+ 4x
–
12
ANSWER:
–
6,
–
2, 1
34.
f (x) = x
3
+ x
2
–
17x + 15
ANSWER:
–
5, 1, 3
35.
f (x) = x
4
–
3x
3
–
3x
2
–
75x
–
700
ANSWER:
–
4, 7,
–
5
i
, 5
i
36.
f (x) = x
4
+ 6x
3
+ 73x
2
+ 384x + 576
ANSWER:
–
3,
–
3,
–
8
i
, 8
i
37.
f (x) = x
4
–
8x
3
+ 20x
2
–
32x + 64
ANSWER:
4, 4,
–
2
i
, 2
i
38.
f (x) = x
5
–
8x
3
–
9x
ANSWER:
–
3, 0, 3,
–
i
,
i
Write a polynomial function of least degree with
integral coefficients that have the given zeros.
39.
5,
–
2,
–
1
ANSWER:
x
3
–
2x
2
–
13x
–
10
40.–
4,
–
3, 5
ANSWER:
x
3
+ 2x
2
–
23x
–
60
41.–
1,
–
1, 2
i
ANSWER:
x
4
+ 2x
3
+ 5x
2
+ 8x + 4
42.–
3, 1,
–
3
i
ANSWER:
x
4
+ 2x
3
+ 6x
2
+ 18x
–
27
43.
0,
–
5, 3 +
i
ANSWER:
x
4
–
x
3
–
20x
2
+ 50x
44.–
2,
–
3, 4
–
3
i
ANSWER:
x
4
–
3x
3
–
9x
2
+ 77x + 150
45.
CCSS REASONING
A computer manufacturer
determines that for each employee, the profit for
producing x computers per day is P(x) =
–
0.006x
4
+
0.15x
3
–
0.05x
2
–
1.8x.
a.
How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b.
What is the meaning of the zeros in this situation?
ANSWER:
a.
2 or 0; 1; 1 or 3
b.
Nonnegative roots represent numbers of
computers produced per day which lead to no profit
for the manufacturer.
Sketch the graph of each function using its
zeros.
46.
ANSWER:
47.
ANSWER:
48.
ANSWER:
49.
ANSWER:
Match each graph to the given zeros.
a.
–
3, 4, i,
–
i
b.
–
4, 3
c.
–
4, 3, i,
–
i
50.
ANSWER:
c
51.
ANSWER:
b
52.
ANSWER:
a
53.
CONCERTS
The amount of money Hoshi
’
s Music
Hall took in from 2003 to 2010 can be modeled by M
(x) =
–
2.03x
3
+ 50.1x
2
–
214x + 4020, where x is the
years since 2003.
a. How many positive real zeros, negative real zeros,
and imaginary zeros exist?
b. Graph the function using your calculator.
c. Approximate all real zeros to the nearest tenth.
What is the significance of each zero in the context
of the situation?
ANSWER:
a.
3 or 1; 0; 2 or 0
b.
[
–
10, 40] scl: 5 by [
–
4000, 13,200]
scl: 100
c.
23.8; Sample answer: According to the model, the
music hall will not earn any money after 2026.
Determine the number of positive real zeros,
negative real zeros, and imaginary zeros for
each function. Explain your reasoning.
54.
degree: 3
ANSWER:
0 positive, 1 negative, 2 imaginary;
Sample answer: The graph does not cross the
positive x-axis, and crosses the negative x-axis once.
Because the degree of the polynomial is 3, there are
3
–
1 or 2 imaginary zeros.
55.
degree:5
ANSWER:
1 positive, 2 negative, 2 imaginary;
Sample answer: The graph crosses the positive x-
axis once, and crosses the negative x-axis twice.
Because the degree of the polynomial is 5, there are
5
–
3 or 2 imaginary zeros.
56.
OPEN ENDED
Sketch the graph of a polynomial
function with:
a.
3 real, 2 imaginary zeros
b.
4 real zeros
c.
2 imaginary zeros
ANSWER:
a.
b.
c.
57.
CHALLENGE
Write an equation in factored form
of a polynomial function of degree 5 with 2 imaginary
zeros, 1 non integral zero, and 2 irrational zeros.
Explain.
ANSWER:
Sample answer:
Use
conjugates for the imaginary and irrational values.
58.
CCSS ARGUMENTS
Determine which equation is
not like the others, Explain
ANSWER:
r
4
+ 1 = 0; Sample answer: The equation has
imaginary solutions and all of the others have real
solutions.
59.
REASONING
Provide a counter example for each
statement.
a.
All polynomial functions of degree greater than 2
have at least 1 negative real root.
b.
All polynomial functions of degree greater than 2
have at least 1 positive real root.
ANSWER:
a.
Sample answer: f (x) = x
4
+ 4x
2
+ 4
b.
Sample answer: f (x) = x
3
+ 6x
2
+ 9x
60.
WRITING IN MATH
Explain to a friend how you
would use Descartes
’
Rule of Signs to determine the
number of possible positive real roots and the number
of possible negative roots of the polynomial function f
(x) = x
4
–
2x
3
+ 6x
2
+ 5x
–
12.
ANSWER:
Sample answer: To determine the number of positive
real roots, determine how many time the signs
change in the polynomial as you move from left to
right. In this function there are 3 changes in sign.
Therefore, there may be 3 or 1 positive real roots. To
determine the number of negative real roots, I would
first evaluate the polynomial for
–
x. All of the terms
with an odd-degree variable would change signs.
Then I would again count the number of sign
changes as I move from left to right. There would be
only one change. Therefore there may be 1 negative
root.
61.
Use the graph of the polynomial function below.
Which is not a factor of the polynomial x
5
+ x
4
–
3x
3
–
3x
2
–
4x
–
4?
A
x
–
2
B
x + 2
C
x
–
1
D
x + 1
ANSWER:
C
62.
SHORT RESPONSE
A window is in the shape of
an equilateral triangle. Each side of the triangle is 8
feet long. The window is divided in half by a support
from one vertex to the midpoint of the side of the
triangle opposite the vertex. Approximately how long
is the support?
ANSWER:
6.9 feet
63.
GEOMETRY
In rectangle ABCD, is 8 units
long. What is the length of ?
F
4units
G
8 units
H
units
J
16 units
ANSWER:
H
64.
SAT/ACT
The total area of a rectangle is 25a
4
–
16b
2
square units. Which factors could represent the
length and width?
A
(5a
2
+ 4b) units and (5a
2
+ 4b) units
B
(5a
2
+ 4b) units and (5a
2
–
4b) units
C
(5a
2
–
4b) units and (5a
2
–
4b) units
D
(5a
–
4b) units and (5a
–
4b) units
E
(5a + 4b) units and (5a
–
4b) units
ANSWER:
B
Use synthetic substitution to find f (
–
8) and f (4)
for each function.
65.
f (x) = 4x
3
+ 6x
2
–
3x + 2
ANSWER:
f(
–
8) =
–
1638; f (4) = 342
66.
f (x) = 5x
4
–
2x
3
+ 4x
2
–
6x
ANSWER:
f(
–
8) = 21,808; f (4) = 1192
67.
f (x) = 2x
5
–
3x
3
+ x
2
–
4
ANSWER:
f(
–
8) =
–
63,940; f (4) = 1868
Factor completely. If the polynomial is not
factorable, write prime.
68.
x
6
–
y
6
ANSWER:
69.
a
6
+ b
6
ANSWER:
(a
2
+ b
2
)(a
4
–
a
2
b
2
+ b
4
)
70.
4x
2
y + 8xy + 16y
–
3x
2
z
–
6xz
–
12z
ANSWER:
(x
2
+ 2x + 4)(4y
–
3z)
71.
5a
3
–
30a
2
+ 40a + 2a
2
b
–
12ab + 16b
ANSWER:
(a
–
4)(a
–
2)(5a + 2b)
72.
BUSINESS
A mall owner has determined that the
relationship between monthly rent charged for store
space r (in dollars per square foot) and monthly profit
P(r) (in thousands of dollars) can be approximated by
P(r) =
–
8.1r
2
+ 46.9r
–
38.2. Solve each quadratic
equation or inequality. Explain what each answer
tells about the relationship between monthly rent and
profit for this mall.
a.
–
8.1r
2
+ 46.9r
–
38.2 = 0
b
.
–
8.1r
2
+ 46.9r
–
38.2 > 0
c.
–
8.1r
2
+ 46.9r
–
38.2 > 10
d.
–
8.1r
2
+ 46.9r
–
38.2 < 10
ANSWER:
a.
0.98, 4.81; The owner will break even if he
charges $0.98 or $4.81 per square foot.
b.
0.98 < r < 4.81; The owner will make a profit if
the rent per square foot is between $0.98 and $4.81.
c.
1.34 < r < 4.45; If rent is set between $1.34 and
$4.45 per sq ft, the profit will be greater than
$10,000.
d.
r < 1.34 or r > 4.45; If rent is set between $0 and
$1.34 or above $4.45 per sq ft, the profit will be less
than $10,000.
73.
DIVING
To avoid hitting any rocks below, a cliff
diver jumps up and out. The equation h =
–
16t
2
+ 4t
+ 26 describes her height h in feet t seconds after
jumping. Find the time at which she returns to a
height of 26 feet.
ANSWER:
0.25 s
Findallofthepossiblevaluesof foreach
replacement set.
74.
a
= {1, 2, 4}; b = {1, 2, 3, 6}
ANSWER:
±
1,
±
2,
±
3,
±
6,
75.
a
= {1, 5}; b = {1, 2, 4, 8}
ANSWER:
76.
a
= {1, 2, 3, 6}; b = {1, 7}
ANSWER:
eSolutionsManual-PoweredbyCogneroPage8
5-7 Roots and Zeros
