Factoring Polynomials

Before factoring any polynomial, write the polynomial indescending order of one of the variables. Then note how manyterms there are, and proceed by using one or more of thefollowing techniques.

1. ALWAYS Factor out the Greatest CommonFactor (GCF) first. Look for this in every problem. This includesfactoring 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, andfactor out the greatest common factor from each group.

Example:

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

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

(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) factorsinto conjugate binomials (conjugate means terms are separated bya plus sign in one binomial and a minus sign in the otherbinomial):

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+64ydoes not factor because it is the SUM of squares

c. The sum of cubes (SOC) or difference ofcubes (DOC) factors by these patterns: (each type contains abinomial 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 amultiple of three