analyzing and solving polynomial equations

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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)  x4 − 5 x

2 − 36 = 0 2)  x

3 + 3 x

2 − 14 x − 20 = 0

3)  x3 − 2 x

2 + 3 x − 6 = 0 4)  x

4 − 14 x

2 + 45 = 0

5)  x4 + 6 x

2 + 8 = 0 6)  x

4 + 3 x

2 − 18 = 0

7)  x3 − 1 = 0 8)  x

3 + 3 x

2 − x − 3 = 0

-1-

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9)  x3 − 2 x

2 − 3 x + 6 = 0 10)  x

6 − 2 x

4 − 4 x

2 + 8 = 0

11)  x5 + 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)  x6 − x

4 − x

2 + 1 = 0 14)  x

8 − 26 x

4 + 25 = 0

-2-

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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)  x4 − 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)  x3 + 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)  x3 − 2 x

2 + 3 x − 6 = 0

# of complex roots: 3Possible # 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)  x4 − 14 x

2 + 45 = 0

# of complex roots: 4Possible # 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)  x4 + 6 x

2 + 8 = 0

# of complex roots: 4Possible # 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)  x4 + 3 x

2 − 18 = 0

# of complex roots: 4Possible # 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)  x3 − 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)  x3 + 3 x

2 − x − 3 = 0

# of complex roots: 3Possible # 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-

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9)  x3 − 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)  x6 − 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)  x5 + 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)  x6 + 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)  x6 − 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)  x8 − 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

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