1

FACTORING POLYNOMIALS

Factoring refers to the process of expressing an algebraic expression as a product of prime factors. A factor

is considered prime if it could no longer be expressed further as a product of simpler algebraic expressions, i.e.

its only factors are 1 and itself. An algebraic expression that is expressed as a product of irreducible factors is

factored completely.

Types of Factoring

1. Common Monomial Factor

ax + ay = a (x + y)

2. Difference of Two Squares

x2 − y2 = (x + y) (x− y)

3. Perfect Square Trinomial

x2 + 2xy + y2 = (x + y)2

x2 − 2xy + y2 = (x− y)2

4. Sum and Difference of Two Cubes

x3 + y3 = (x + y) (x2 − xy + y2)

x3 − y3 = (x− y) (x2 + xy + y2)

5. Quadratic Trinomials

x2 + (a + b) x + ab = (x + a) (x + b)

acx2 + (ad + bc) x + bd = (ax + b) (cx + d)

6. Completing the Square

This technique is applicable to polynomials that may be converted to a perfect square trinomial upon addition

and then subtraction of a perfect square term.

7. Factoring by Grouping

This technique is applicable to polynomials that are longer or more complicated. The key lies in grouping

the terms in such a way that the groups have common factor. This may entail several trials before the desired

grouping is arrived at.

EXERCISES

1. −12xy3z2 − 28y3z − 20x2y2z2

2. 4ab3 − 16a3b

3. 4 (x− 1)2 − 9y2

4. (2x− y)3 − 8

5. (x + 3)3 + (y − 1)3

6. 16a2b4 + 40ab2c + 25c2

7. (2s− 3t)2 − 8 (2s− 3t) + 16

8. 5x3 − 10x2y − 75xy2

9. x (x + 1) (4x− 5)− 6 (x + 1)

10. a2 − ab + a− b

11. x2 + xy − 2y2 + 2x− 2y

12. 4x2 − y2 + 2yz − z2

13. 9x2 − 12xy + 4y2 − 25z2 + 10zw − w2

14. x4 + 64

15. x4 − 11x2 + 1

16. 16x4 − 24x2y2 + 25y2

mong!