WORKSHEET ON FACTORING POLYNOMIALS

Factor each of the following.

Problem 1 :

x² - 9

Problem 2 :

x³ + 8

Problem 3 :

x³ - 1

Problem 4 :

9x² - 16

Problem 5 :

x⁴ - y⁴

Problem 6 :

a²b - b³

Problem 7 :

3x⁵ + 21x⁴ - 3x³ - 21x²

Problem 8 :

2x⁴ - 5x² - 12

In problems 9, 10, 11 and 12, P = 5 and Q = 13.

Problem 9 :

x² + Px + Qx + PQ

Problem 10 :

x³ - Px² + 2x² - 4x + 4P - 8

Problem 11 :

x³ - 4x² - Px² - 5x + 4Px + 5P

Problem 12 :

x³ + P³

Answers

1. Answer :

= x² - 9

= x² - 3²

Use the algebraic identity a² - b² = (a + b)(a - b).

= (x + 3)(x - 3)

2. Answer :

= x² + 8

= x² + 2³

Use the algebraic identity a³ + b³ = (a + b)(a² - ab + b²).

= (x + 2)(x² - x ⋅ 2 + 2²)

= (x + 2)(x² - 2x + 4)

3. Answer :

= x² - 1

= x² - 1³

Use the algebraic identity a³ - b³ = (a - b)(a² + ab + b²).

= (x - 1)(x² - x ⋅ 1 + 1²)

= (x - 1)(x² - x + 1)

4. Answer :

= 9x² - 16

= 3²x² - 4²

= (3x)² - 4²

= (3x + 4)(3x - 4)

5. Answer :

= x⁴ - y⁴

= (x²)² - (2²)²

= (x² + 2²)(x² - 2²)

= (x² + 4)(x + 2)(x - 2)

6. Answer :

= a²b - b³

= b(a² - b²)

= b(a + b)(a - b)

7. Answer :

= 3x⁵ + 21x⁴ - 3x³ - 21x²

= 3x⁴(x + 7) - 3x²(x + 7)

= (3x⁴ - 3x²)(x + 7)

= 3x²(x² - 1)(x + 7)

= 3x²(x² - 1²)(x + 7)

= 3x²(x + 1)(x - 1)(x + 7)

8. Answer :

= 2x⁴ - 5x² - 12

= 2(x²)² - 5x² - 12

Let y = x².

= 2y² - 5y - 12

= 2y² - 8y + 3y - 12

= = 2y(y - 4) +3(y - 4)

= (2y + 3)(y - 4)

Replace y by x².

= (2x² + 3)(x² - 4)

= (2x² + 3)(x² - 2²)

= (2x² + 3)(x + 2)(x - 2)

9. Answer :

10. Answer :

11. Answer :

12. Answer :

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