Factoring Polynomials Worksheet

Solution of exercise 1

x³ + x²

x³ + x² = x² (x + 1)

x = 0, x = −1

Solution of exercise 2

2x⁴ + 4x² = 2x² (x² + 2)

2x⁴ + 4x² = 2x² (x² + 2)

x = 0

Solution of exercise 3

x² − 4

x² − 4 = (X + 2) · (X − 2)

x = −2, X = 2

Solution of exercise 4

x⁴ − 16

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

x = −2 , X = 2

Solution of exercise 5

9 + 6x + x²

x = −3

Solution of exercise 6

x = 3, x = −2.

Solution of exercise 7

x⁴ − 10x² + 9

x² = t

x⁴ − 10x² + 9 = 0

t² − 10t + 9 = 0

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

Solution of exercise 8

x⁴ − 2x² − 3

x² = t

t² − 2t − 3 = 0

x⁴ − 2x² + 3 = (x² + 1) · (x + )

Solution of exercise 9

2x⁴ + x³ − 8x² − x + 6

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

(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

(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

(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

Solution of exercise 10

2x³ − 7x² + 8x − 3

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

(x −1) · (2x² − 5x + 3)

P(1) = 2 · 1 ² −5 · 1 + 3 = 0

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

x = 3/2, x = 1

Solution of exercise 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

(x − 2) · (x² + x + 2 )

x²+ x + 2 = 0

(x − 2) · (x² + x + 2 )

x = 2

Solution of exercise 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

(x − 2) · (x² + 5x + 6)

x² + 5x + 6 = 0

(x − 2) · (x + 2) · (x +3)

x = 2, x = −2, x = −3.

Solution of exercise 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

(x+2) · (6x² − 5x + 1)

6x² − 5x + 1 = 0

6 · (x + 2) · (x − 1/2) · (x − 1/3)

x = − 2, x = 1/2, x= 1/3

Solution of exercise 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

2x² + x − 28 = 2 (x + 4) · (x − 7/2)

Solution of exercise 15

1

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

x² − 11x + 30 = (x −6) · (x −5)

133x² + 10x + 3

3x² + 10x + 3 = 0

3x² + 10x + 3 = 3 (x − 3) · (x − 1/3)

142x² − x − 1

2x² − x −1 = 0

2x² − x − 1 = 2 (x − 1) · (x + 1/2)