Factoring Polynomials

Before factoring any polynomial, write the polynomial in descending order of one of the variables. Then note how many terms there are, and proceed by using one or more of the following techniques.

1. If there are THREE TERMS , look for these patterns:

a. Quadratic trinomials of the form ax + bx + c where a = 1 (QT a = 1) factor into the product of two binomials (double bubble) where the factors of c must add to b.

Example: x - 4x - 12 = (x - 6)(x + 2)

b. Quadratic trinomials of the form ax + bx + c where a 1 (QT a 1) eventually factor into the product of two binomials (double bubble) but you must first find the factors of ac that add to b, rewrite the original replacing b with these factors of ac, then factor by grouping to finally get to the double bubble.

Example: 9x + 15x + 4

ac = (9)(4) = 36

factors of 36 that add to 15: 12 and 3

9x + 12x + 3x + 4 rewrite 15x as 12x + 3x, then factor by grouping 3x(x + 4) + 1(x + 4) = (x + 4)(3x + 1)

c. Quadratic square trinomials (QST) of the form ax + bx + c may factor into the square of a binomial. Look for the pattern where two of the terms are perfect squares, and the remaining term is twice the product of the square root of the squares:

a ± 2ab + b = (a ± b)

Example: 16x - 40x + 25 = (4x - 5)

Note the pattern: Square root of 16 x is 4 x. Square root of 25 is 5.

Twice the product of the square roots: 2(4x)(5) = 40x, which is the middle term

2. Factor all expressions completely. Sometimes, you will need to use two or three types of factoring in a single problem.

Example: -2x + 32

-2x + 32 = factor out the GCF of -2x and 32

-2(x - 16) = factor the difference of squares

-2(x - 4)(x + 4) = factor the remaining difference of squares

-2(x - 2)(x + 2)(x + 4) (remember that the sum of squares is prime)