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. ALWAYS Factor out the Greatest Common Factor (GCF) first. Look for this in every problem. This includes factoring out the negative sign if it precedes the leading term.

Example: -x + 6x - 3 = -1(x - 6x + 3)

Example: 4xy-8xy= 4xy(xy - 2) where 4xy was the GCF.

2. If there are FOUR TERMS , try to factor by grouping (GR). Group two terms at a time, and factor out the greatest common factor from each group.

Example:

x+ 6x- 2x - 12 = group the first two terms then the last two terms

x(x + 6) -2(x + 6) = factor the (x + 6) out of both terms

(x + 6)(x - 2) this is the factored answer

3. If there are TWO TERMS , look for one of these patterns:

a. The difference of squares (DOS) factors into conjugate binomials (conjugate means terms are separated by a plus sign in one binomial and a minus sign in the other binomial):

a - b = (a - b)(a + b)

Example: 9x-64y= (3x-8y)(3x+8y)

Note: a variable is a perfect square if the exponent is even

b. The sum of squares does not factor: a + b is prime (doesn't factor)

Example: 9x+64y does not factor because it is the SUM of squares

c. The sum of cubes (SOC) or difference of cubes (DOC) factors by these patterns: (each type contains a binomial times a trinomial)

a + b= (a + b)(a -ab + b)

Example: 8x + 27 = (2x + 3)(4x -6x + 9)

a - b= (a- b)(a + ab + b)

Example: 64x - 125y= (4x - 5y)(16x + 20x + 25y)

Note: a variable is a perfect cube if the exponent is a multiple of three