Summary of Factoring Techniques

  • For all polynomials, first factor out the greatest common factor (GCF).

  • For a binomial, check to see if it is any of the following:

    1. difference of squares: x 2y 2 = ( x + y) ( xy)

    2. difference of cubes: x 3y 3 = ( xy) ( x 2 + xy + y 2)

    3. sum of cubes: x 3 + y 3 = ( x + y) ( x 2xy + y 2)

  • For a trinomial, check to see whether it is either of the following forms:

    1. x 2 + bx + c:

      If so, find two integers whose product is c and whose sum is b. For example,

      x 2 + 8 x + 12 = ( x + 2)( x + 6)

      since (2)(6) = 12 and 2 + 6 = 8

    2. ax 2 + bx + c:

      If so, find two binomials so that

      • the product of first terms = ax 2

      • the product of last terms = c

      • the sum of outer and inner products = bx

      See the following polynomial in which the product of the first terms = (3 x)(2 x) = 6 x 2, the product of last terms = (2)(–5) = –10, and the sum of outer and inner products = (3 x)(–5) + 2(2 x) = –11 x.

      equation

    3. equation

  • For polynomials with four or more terms, regroup, factor each group, and then find a pattern as in steps 1 through 3.