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How to Factor the Difference of Cubes When the Exponent Isn't 3

The difference of cubes provides a quick method of factoring a binomial consisting of two numbers that have whole numbers and variables as the cube roots. The difference of cubes factors as (x^3) -- (a^3) = (x - a)(x^2 + ax + a^2). Because of the equation, educators typically teach students to recognize a difference of cubes through the number "3" as one or both of the exponents. However, the exponents do not have to be "3" for the term to be a cube, but rather the exponent must be divisible by 3.

Instructions

    • 1

      Remove any common factors needed to make the binomial into a difference of cubes, if applicable. For example, with (27x^13) -- (125x), factor out one x, because x [(27x^12) -- 125] has a difference of cubes.

    • 2

      Find the cube root of each coefficient through memorization or a calculator, and divide the exponents by 3. For example, x [(27x^12) -- 125] becomes x {[(3x^4)^3] -- (5)^3}.

    • 3

      Fill in the equation for the difference of cubes. For the example, fill in (x - a)(x^2 + ax + a^2) to make x [(3x^4) -- 5] {[(3x^4)^2] + [5 * (3x^4)] + (5)^2}.

    • 4

      Simplify by multiplying coefficients and exponents where applicable. The factor for the example simplifies to x [(3x^4) -- 5] [(9x^8) + (15x^4) + 25].

    • 5

      Multiply the solution to check your work, if desired.

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