Factoring Polynomials Worksheet

Solutions

Factor and Calculate the Roots of the Following Polynomials

1 x3 + x2

22x4 + 4x2

3x2 − 4

4x4 − 16

59 + 6x + x2

6trinomio

7x4 − 10x2 + 9

8x4 − 2x2 3

92x4 + x3 − 8x2 − x + 6

102x3 − 7x2 + 8x − 3

11x3 − x2 − 4

12x3 + 3x2 − 4 x − 12

136x3 + 7x2 − 9x + 2

14Factor:

19x4 − 4x2 =

2x5 + 20x3 + 100x =

33x5 − 18x3 + 27x =

42x3 − 50x =

52x5 − 32x =

62x2 + x − 28 =

15Factor:

1Factored Polynomial Exercise

2xy − 2x − 3y + 6 =

325x2 − 1=

436x6 − 49 =

5x2 − 2x + 1 =

6x2 − 6x + 9 =

7x2 − 20x + 100 =

8x2 + 10x +25 =

9x2 + 14x + 49 =

10x3 − 4x2 + 4x =

113x7 − 27x =

12x2 − 11x + 30

133x2 + 10x + 3

142x2 − x − 1

1

x3 + x2

x3 + x2 = x2 (x + 1)

x = 0, x = −1


2

2x4 + 4x2 = 2x2 (x2 + 2)

2x4 + 4x2 = 2x2 (x2 + 2)

x = 0


3

x2 − 4

x2 − 4 = (X + 2) · (X − 2)

x = −2, X = 2


4

x4 − 16

x4 − 16 = (x2 + 4) · (x2 − 4) = (X + 2) · (X − 2) · (x2 + 4)

x = −2 , X = 2


5

9 + 6x + x2

Factored Polynomial Operations

x = −3


6

Factored Polynomial Exercise

Factored Polynomial Exercise

Factored Polynomial Operations

Factored Polynomial Operations

x = 3, x = −2.


7

x4 − 10x2 + 9

x2 = t

x4 − 10x2 + 9 = 0

t2 − 10t + 9 = 0

Factored Polynomial Operations

soluciones

soluciones

x4 − 10x2 + 9 = (x + 1) · (x − 1) · (x + 3) · (x − 3)


8

x4 − 2x2 3

x2 = t

t2 − 2t − 3 = 0

Factored Polynomial Operations

soluciones

soluciones

x4 − 2x2 + 3 = (x2 + 1) · (x + RAÍZ DE TRES)


9

2x4 + x3 − 8x2 − x + 6

P(1) = 2 · 14 + 13 − 8 · 12 − 1 + 6 = 2 + 1− 8 − 1 + 6 = 0

Factored Polynomial Operations

(x −1) · (2x3 + 3x2 − 5x − 6 )

P(1) = 2 · 13 + 3 · 12 − 5 · 1 − 6≠ 0

P(−1) = 2 · (− 1)3 + 3 ·(− 1)2 − 5 · (− 1) − 6= −2 + 3 + 5 − 6 = 0

Factored Polynomial Operations

(x −1) · (x +1) · (2x2 +x −6)

P(−1) = 2 · (−1)2 + (−1) − 6 ≠ 0

P(2) = 2 · 22 + 2 − 6 ≠ 0

P(−2) = 2 · (−2)2 + (−2) − 6 = 2 · 4 − 2 − 6 = 0

Factored Polynomial Operations

(x −1) · (x +1) · (x +2) · (2x −3 )

2x −3 = 2 (x − 3/2)

2x4 + x3 − 8x2 − x + 6 = 2 (x −1) · (x +1) · (x +2) · (x − 3/2)

x = 1, x = − 1, x = −2, x = 3/2


10

2x3 − 7x2 + 8x − 3

P(1) = 2 · 13 − 7 · 12 + 8 · 1 − 3 = 0

Factored Polynomial Operations

(x −1) · (2x2 − 5x + 3)

P(1) = 2 · 1 2 −5 · 1 + 3 = 0

Factored Polynomial Operations

(x −1)2 · (2x −3) = 2 (x − 3/2) · (x −1)2

x = 3/2, x = 1


11

x3 − x2 − 4

{±1, ±2, ±4 }

P(1) = 1 3 − 1 2 − 4 ≠ 0

P(−1) = (−1) 3 − (−1) 2 − 4 ≠ 0

P(2) = 2 3 − 2 2 − 4 = 8 − 4 − 4 = 0

Factored Polynomial Operations

(x − 2) · (x2 + x + 2 )

x2 + x + 2 = 0

Factored Polynomial Operations

(x − 2) · (x2 + x + 2 )

x = 2


12

x3 + 3x2 − 4x − 12

{±1, ±2, ±3, ±4, ±6, ±12 }

P(1) = 13 + 3 · 12 − 4 · 1 − 12 ≠ 0

P(−1) = (−1)3 + 3 · (−1)2 − 4 · (−1) − 12 ≠ 0

P(2) = 23 + 3 · 22 − 4 · 2 − 12 = 8 + 12 − 8 − 12 = 0

Factored Polynomial Operations

(x − 2) · (x2 + 5x + 6)

x2 + 5x + 6 = 0

Factored Polynomial Operations

(x − 2) · (x + 2) · (x +3)

x = 2, x = −2, x = −3.


13

6x3 + 7x2 − 9x + 2

{±1, ±2}

P(1) = 6 · 13 + 7 · 12 − 9 · 1 + 2 ≠ 0

P(−1) = 6 · (−1)3 + 7 · (−1)2 − 9 · (−1) + 2 ≠ 0

P(2) = 6 · 2 3 + 7 · 2 2 − 9 · 2 + 2 ≠ 0

P(−2) = 6 · (−2)3 + 7 · (−2)2 − 9 · (−2) + 2 = − 48 + 28 + 18 + 2 = 0

Factored Polynomial Operations

(x+2) · (6x2 − 5x + 1)

6x2 − 5x + 1 = 0

Factored Polynomial Operations

6 · (x + 2) · (x − 1/2) · (x − 1/3)

x = − 2, x = 1/2, x= 1/3


14

19x4 − 4x2 =

x2 · (9x2 − 4) =

x2 · (3x + 2) · (3x − 2)

2x5 + 20x3 + 100x =

x · (x4 + 20x2 + 100) =

x · (x2 + 10)2

33x5 − 18x3 + 27x =

3x · (x4 − 6x2 + 9) =

= 3x · (x2 − 3)2

42x3 − 50x =

=2x · (x2 − 25) =

2x · (x + 5) · (x - 5)

52x5 − 32x =

= 2x · (x4 − 16 ) =

2x · (x2 + 4) · (x2 − 4) =

= 2x · (x2 + 4) ·(x +2) · (x − 2)

62x2 + x − 28

2x2 + x − 28 = 0

Factored Polynomial Operations

2x2 + x − 28 = 2 (x + 4) · (x − 7/2)


15

1Factored Polynomial Exercise

Factored Polynomial Solution

2xy − 2x − 3y + 6 =

= x · (y − 2) − 3 · (y − 2) =

= (x − 3) · (y − 2)

325x2 − 1=

= (5x +1) ·(5x − 1)

436x6 − 49 =

= (6x3 + 7) · (6x3 − 7)

5x2 − 2x + 1 =

= (x − 1)2

6x2 − 6x + 9 =

= (x − 3)2

7x2 − 20x + 100 =

= (x − 10)2

8x2 + 10x + 25 =

= (x + 5)2

9x2 + 14x + 49 =

= (x + 7)2

10x3 − 4x2 + 4x =

= x · (x2 − 4x +4) =

= x · (x − 2)2

113x7 − 27x =

= 3x · (x6 − 9) =

= 3x · (x3 + 3) · (x3 − 3)

12x2 − 11x + 30

x2 − 11x + 30 = 0

Factored Polynomial Operations

x2 − 11x + 30 = (x −6) · (x −5)

133x2 + 10x + 3

3x2 + 10x + 3 = 0

Factored Polynomial Operations

3x2 + 10x + 3 = 3 (x − 3) · (x − 1/3)

142x2 − x − 1

2x2 − x −1 = 0

Factored Polynomial Operations

2x2 − x − 1 = 2 (x − 1) · (x + 1/2)