7.6 Polynomials and Factoring Part 2: Factoring

Factoring The process of finding polynomials whose product equals a given polynomial is called factoring. For example, 4(x + 3) is the factored form of 4x + 12 (4 was “factored out” since it was a factor of both terms). A polynomial that cannot be “factored” is called a prime polynomial. A polynomial is factored completely when it is written as the product of prime polynomials.

Factoring Out the Greatest Common Factor Some polynomials are factored by using the [reverse] Distributive Property. We look for a monomial that is the GCF of all terms in the polynomial. 6x 2 y 3 + 9xy y 5

 Factor out the greatest common factor from each polynomial. 9y 5 + y 2 6x 2 t + 8xt + 12t 14m 4 (m + 1) – 28m 3 (m – 1) – 7m 2 (m – 1)

Factoring by Grouping When a polynomial has more than three terms, it can sometimes be factored by a method called factoring by grouping. ax + ay + 6x + 6y

 Factor by grouping. mp 2 + 7m + 3p y 2 – 2z – ay 2 + az

Factoring Trinomials (Guess & Check Method) Factor each trinomial. 4y 2 – 11y + 6 6p 2 – 7p – 5

 Factor each trinomial. x 2 – 2x – 15 6a 2 – 48a – 120

Factoring Perfect Square Trinomials Perfect Square Trinomials x 2 + 2xy + y 2 = (x + y) 2 x 2 – 2xy + y 2 = (x – y) 2 Factor the polynomial. 16p 2 – 40pq + 25q 2 If you don’t realize you’re factoring a perfect square trinomial, you can still use Guess & Check!

 Factor the polynomial. 9m 2 – 12m + 4 4x 2 y xy + 49

Factoring Special Binomials Difference of Squares x 2 – y 2 = (x + y)(x – y) Sum of Cubes x 3 + y 3 = (x + y)(x 2 – xy + y 2 ) Difference of Cubes x 3 – y 3 = (x – y)(x 2 + xy + y 2 )

Factoring Special Binomials Factor each polynomial. 4m 2 – 9 x

 Factor each polynomial. m 3 – 64n 3 256k 4 – 625m 4 8q p 9