Irrational and Imaginary Root Theorems - Kuta Software

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Oct 8, 2013 (4 years and 2 months ago)

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Worksheet by Kuta Software LLC
Kuta Software - Infinite Algebra 2 Name___________________________________
Period____Date________________
Irrational and Imaginary Root Theorems
State the number of complex zeros and the possible number of real and imaginary zeros for each
function.
1)


f (


x) =



x
2
+ 6


x − 38 2)


f (


x) =



x
4
− 9


x
2
+ 18
3)


f (


x) =

5


x
5
+ 36


x
3
+ 7


x 4)


f (


x) =

3


x
3
+ 2


x
2



x
5)


f (


x) =

27


x
6
+ 208


x
3
− 64 6)


f (


x) =

27


x
9
+ 8


x
6
− 27


x
3
− 8
A polynomial function with rational coefficients has the follow zeros. Find all additional zeros.
7) −5, i
8)

−1 + i, 5
9)

−3 + 5
, −i 10) 2,

−2 + 10
11) −1, 5,

−2 + 5
12)

2 − 2i,

1 − 2i,

1 + 2i
-1-
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Worksheet by Kuta Software LLC
13)

2 − 3i mult. 2
14)


52
, 5
mult. 2
Write a polynomial function of least degree with integral coefficients that has the given zeros.
15) 0, 2, 3
16) −5, 3
17) −1, 2i 18) 2i, −2i,

2 + 2i
19) −2i,

2 +

2 2
20) 6
,

−3 + 5
Critical thinking questions:
21) Write a polynomial function of fifth degree
with integral coefficients that has 2i as a zero.
22) True or False: A polynomial function of third
degree with integral coefficients can have 2
and 2i as zeros.
-2-
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Algebra 2 Name___________________________________
Period____Date________________
Irrational and Imaginary Root Theorems
State the number of complex zeros and the possible number of real and imaginary zeros for each
function.
1)


f (


x) =



x
2
+ 6


x − 38
# of complex zeros: 2
Possible # of real zeros: 2 or 0
Possible # of imaginary zeros: 2 or 0
2)


f (


x) =



x
4
− 9


x
2
+ 18
# of complex zeros: 4
Possible # of real zeros: 4, 2, or 0
Possible # of imaginary zeros: 4, 2, or 0
3)


f (


x) =

5


x
5
+ 36


x
3
+ 7


x
# of complex zeros: 5
Possible # of real zeros: 5, 3, or 1
Possible # of imaginary zeros: 4, 2, or 0
4)


f (


x) =

3


x
3
+ 2


x
2



x
# of complex zeros: 3
Possible # of real zeros: 3 or 1
Possible # of imaginary zeros: 2 or 0
5)


f (


x) =

27


x
6
+ 208


x
3
− 64
# of complex zeros: 6
Possible # of real zeros: 6, 4, 2, or 0
Possible # of imaginary zeros: 6, 4, 2, or 0
6)


f (


x) =

27


x
9
+ 8


x
6
− 27


x
3
− 8
# of complex zeros: 9
Possible # of real zeros: 9, 7, 5, 3, or 1
Possible # of imaginary zeros: 8, 6, 4, 2, or 0
A polynomial function with rational coefficients has the follow zeros. Find all additional zeros.
7) −5, i
−i
8)

−1 + i, 5

−1 − i,

− 5
9)

−3 + 5
, −i

−3 −

5
, i
10) 2,

−2 + 10

−2 −

10
11) −1, 5,

−2 + 5

−2 −

5
12)

2 − 2i,

1 − 2i,

1 + 2i

2 + 2i
-1-
©
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Worksheet by Kuta Software LLC
13)

2 − 3i mult. 2

2 + 3i mult. 2
14)


52
, 5
mult. 2

− 5
mult. 2
Write a polynomial function of least degree with integral coefficients that has the given zeros.
15) 0, 2, 3


f (


x) =



x
4
− 2


x
3
− 3


x
2
+ 6


x
16) −5, 3


f (


x) =



x
3
+ 5


x
2
− 3


x − 15
17) −1, 2i


f (


x) =



x
3
+


x
2
+ 4


x + 4
18) 2i, −2i,

2 + 2i


f (


x) =



x
4
− 4


x
3
+ 12


x
2
− 16


x + 32
19) −2i,

2 +

2 2


f (


x) =



x
4
− 4


x
3
− 16


x − 16
20) 6
,

−3 + 5


f (


x) =



x
4
+ 6


x
3
− 2


x
2
− 36


x − 24
Critical thinking questions:
21) Write a polynomial function of fifth degree
with integral coefficients that has 2i as a zero.
Many answers. Ex:


f (


x) =



x
5
+ 4


x
3
22) True or False: A polynomial function of third
degree with integral coefficients can have 2
and 2i as zeros.
True
-2-
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