©
v
T
2
u
0
6
1
f
2
9
5
K
S
u
H
t
R
a
9
q
S
T
o
4
f
P
t
g
w
I
a
B
r
h
e
o
W
L
L
L
x
C
9
.
8
s
J
A
I
l
E
l
G
K
r
r
i
4
g
B
h
9
t
9
s
q
H
r
p
e
n
s
L
e
y
r
b
v
H
e
D
d
O
.
3
t
5
M
l
a
v
d
G
e
y
Z
w
s
i
a
t
J
h
c
C
I
8
n
b
f
2
i
P
n
Y
i
m
t
s
e
z
V
A
h
l
4
g
J
e
6
b
q
r
c
a
k
i
2
Q
.
U
Worksheet by Kuta Software LLC
Kuta Software - Infinite Algebra 2 Name___________________________________
Period____Date________________
Analyzing and Solving Polynomial Equations
State the number of complex roots, the possible number of real and imaginary roots, the possible
number of positive and negative roots, and the possible rational roots for each equation. Then find all
roots.
1)
x
4
− 5
x
2
− 36 = 0 2)
x
3
+ 3
x
2
− 14
x − 20 = 0
3)
x
3
− 2
x
2
+ 3
x − 6 = 0 4)
x
4
− 14
x
2
+ 45 = 0
5)
x
4
+ 6
x
2
+ 8 = 0 6)
x
4
+ 3
x
2
− 18 = 0
7)
x
3
− 1 = 0 8)
x
3
+ 3
x
2
−
x − 3 = 0
-1-
©
o
N
2
U
0
n
1
E
2
4
l
K
j
u
3
t
9
a
L
B
S
5
o
R
f
e
t
U
w
E
a
x
r
p
e
U
X
L
0
L
J
C
B
.
E
V
C
A
u
l
Z
l
F
b
r
4
i
T
g
C
h
9
t
N
s
U
y
r
P
e
m
s
A
e
B
r
V
v
j
e
K
d
8
.
L
w
E
M
R
a
F
d
g
e
u
T
w
m
i
H
t
0
h
Z
H
I
A
n
x
f
P
i
U
n
x
i
p
t
w
e
e
5
A
u
l
b
g
D
e
l
b
v
r
b
a
e
k
2
g
.
B
Worksheet by Kuta Software LLC
9)
x
3
− 2
x
2
− 3
x + 6 = 0 10)
x
6
− 2
x
4
− 4
x
2
+ 8 = 0
11)
x
5
+ 2
x
4
+ 11
x
3
+ 22
x
2
+ 24
x + 48 = 0 12)
x
6
+ 5
x
4
− 4
x
2
− 20 = 0
13)
x
6
−
x
4
−
x
2
+ 1 = 0 14)
x
8
− 26
x
4
+ 25 = 0
-2-
©
o
9
2
4
0
S
1
Y
2
F
p
K
5
u
T
t
n
a
S
x
S
V
o
O
f
T
t
7
w
t
a
8
r
A
e
h
n
L
z
L
E
C
m
.
3
B
2
A
N
l
7
l
s
i
r
V
i
n
g
G
h
V
t
I
s
w
5
r
V
e
Y
s
8
e
J
r
F
v
6
e
U
d
X
.
Z
V
W
M
k
a
h
d
T
e
O
R
w
z
i
x
t
Z
h
r
I
I
4
n
q
f
Q
i
1
n
D
i
A
t
z
e
W
J
A
V
l
k
g
j
e
t
b
B
r
V
a
w
m
2
a
.
z
Worksheet by Kuta Software LLC
Kuta Software - Infinite Algebra 2 Name___________________________________
Period____Date________________
Analyzing and Solving Polynomial Equations
State the number of complex roots, the possible number of real and imaginary roots, the possible
number of positive and negative roots, and the possible rational roots for each equation. Then find all
roots.
1)
x
4
− 5
x
2
− 36 = 0
# of complex roots: 4
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4, 2, or 0
Possible # positive real roots: 1
Possible # negative real roots: 1
Possible rational roots:
± 1, ± 2, ± 3, ± 4, ± 6, ± 9, ± 12, ± 18, ± 36
Roots:
{
2i, −2i, 3, −3
}
2)
x
3
+ 3
x
2
− 14
x − 20 = 0
# of complex roots: 3
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible # positive real roots: 1
Possible # negative real roots: 2 or 0
Possible rational roots:
± 1, ± 2, ± 4, ± 5, ± 10, ± 20
Roots:
{
−5,
1 + 5,
1 −
5
}
3)
x
3
− 2
x
2
+ 3
x − 6 = 0
# of complex roots: 3
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible # positive real roots: 3 or 1
Possible # negative real roots: 0
Possible rational roots:
± 1, ± 2, ± 3, ± 6
Roots:
{
2,
i 3,
−i 3
}
4)
x
4
− 14
x
2
+ 45 = 0
# of complex roots: 4
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4, 2, or 0
Possible # positive real roots: 2 or 0
Possible # negative real roots: 2 or 0
Possible rational roots:
± 1, ± 3, ± 5, ± 9, ± 15, ± 45
Roots:
{
5,
− 5, 3, −3
}
5)
x
4
+ 6
x
2
+ 8 = 0
# of complex roots: 4
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4, 2, or 0
Possible # positive real roots: 0
Possible # negative real roots: 0
Possible rational roots:
± 1, ± 2, ± 4, ± 8
Roots:
{
2i, −2i,
i 2,
−i 2
}
6)
x
4
+ 3
x
2
− 18 = 0
# of complex roots: 4
Possible # of real roots: 4, 2, or 0
Possible # of imaginary roots: 4, 2, or 0
Possible # positive real roots: 1
Possible # negative real roots: 1
Possible rational roots:
± 1, ± 2, ± 3, ± 6, ± 9, ± 18
Roots:
{
3,
− 3,
i 6,
−i 6
}
7)
x
3
− 1 = 0
# of complex roots: 3
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible # positive real roots: 1
Possible # negative real roots: 0
Possible rational roots:
± 1
Roots:
{
1,
−1 +
i 3
2
,
−1 −
i 3
2
}
8)
x
3
+ 3
x
2
−
x − 3 = 0
# of complex roots: 3
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible # positive real roots: 1
Possible # negative real roots: 2 or 0
Possible rational roots:
± 1, ± 3
Roots:
{
−3, 1, −1
}
-1-
©
K
2
2
1
0
3
1
7
2
w
4
K
V
u
4
t
x
a
Z
m
S
i
o
h
f
Q
t
t
w
E
a
N
r
5
e
m
G
L
s
L
e
C
0
.
j
o
q
A
a
l
k
l
J
7
r
1
i
D
g
I
h
z
t
T
s
x
6
r
5
e
1
s
m
e
b
r
h
v
S
e
Q
d
W
.
1
F
T
M
S
a
1
d
F
e
q
7
w
L
i
w
t
I
h
2
R
I
O
n
g
f
P
i
A
n
3
i
7
t
a
e
4
w
A
I
l
G
g
G
e
M
b
M
r
I
a
b
B
2
D
.
f
Worksheet by Kuta Software LLC
9)
x
3
− 2
x
2
− 3
x + 6 = 0
# of complex roots: 3
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible # positive real roots: 2 or 0
Possible # negative real roots: 1
Possible rational roots:
± 1, ± 2, ± 3, ± 6
Roots:
{
2, 3,
− 3
}
10)
x
6
− 2
x
4
− 4
x
2
+ 8 = 0
# of complex roots: 6
Possible # of real roots: 6, 4, 2, or 0
Possible # of imaginary roots: 6, 4, 2, or 0
Possible # positive real roots: 2 or 0
Possible # negative real roots: 2 or 0
Possible rational roots:
± 1, ± 2, ± 4, ± 8
Roots:
{
2 mult. 2,
− 2 mult. 2,
i 2,
−i 2
}
11)
x
5
+ 2
x
4
+ 11
x
3
+ 22
x
2
+ 24
x + 48 = 0
# of complex roots: 5
Possible # of real roots: 5, 3, or 1
Possible # of imaginary roots: 4, 2, or 0
Possible # positive real roots: 0
Possible # negative real roots: 5, 3, or 1
Possible rational roots:
± 1, ± 2, ± 3, ± 4, ± 6, ± 8, ± 12, ± 16, ± 24, ± 48
Roots:
{
−2,
i 3,
−i 3,
2i 2,
−2i 2
}
12)
x
6
+ 5
x
4
− 4
x
2
− 20 = 0
# of complex roots: 6
Possible # of real roots: 6, 4, 2, or 0
Possible # of imaginary roots: 6, 4, 2, or 0
Possible # positive real roots: 1
Possible # negative real roots: 1
Possible rational roots:
± 1, ± 2, ± 4, ± 5, ± 10, ± 20
Roots:
{
i 5,
−i 5, 2,
− 2,
i 2,
−i 2
}
13)
x
6
−
x
4
−
x
2
+ 1 = 0
# of complex roots: 6
Possible # of real roots: 6, 4, 2, or 0
Possible # of imaginary roots: 6, 4, 2, or 0
Possible # positive real roots: 2 or 0
Possible # negative real roots: 2 or 0
Possible rational roots:
± 1
Roots:
{
1 mult. 2, −1 mult. 2, i, −i
}
14)
x
8
− 26
x
4
+ 25 = 0
# of complex roots: 8
Possible # of real roots: 8, 6, 4, 2, or 0
Possible # of imaginary roots: 8, 6, 4, 2, or 0
Possible # positive real roots: 2 or 0
Possible # negative real roots: 2 or 0
Possible rational roots:
± 1, ± 5, ± 25
Roots:
{
1, −1, i, −i, 5,
− 5,
i 5,
−i 5
}
-2-
Create your own worksheets like this one with
Infinite Algebra 2
. Free trial available at KutaSoftware.com
