How to Factor Cube Functions

Factoring a cubic is easy if you break it down into steps.

Factoring cube functions usually begins with simply factoring out a common factor. Look for a number, variable or a combination that can be factored out of all terms in the function. If you can factor out a variable, then you will have a plain quadratic function that can be factored like any other quadratic. Otherwise, look for either another common factor, a sum of cubes, or a difference of cubes.

Instructions

1. Learn how to factor quadratics before working with cubic equations. Most of the time, you will need to factor a quadratic function as part of factoring a cube one.

2. Factor out the common factor. Often, there will be either a number, variable or both that is common to all the terms and can be factored out. For example, the cube function 4x^3 + 4 has 4 as a common factor. Therefore, the first step is:

   4x^3 + 4 =

   4(x^3 + 1)

   The function 3x^3 + x has x as a common factor, so the first step is:

   3x^3 + x =

   x(3x^2 +1)

   Sometimes both a number and a variable are common to multiple terms as in:

   6x^3 + 3x^2 + 9x =

   3x(2x^2 + x + 3)

3. Look for a common pattern for a cubic function. Many cube expressions are either a sum or a difference of cubes. Both consist of a cubed variable and a cubed number, but "sum" means they are added and "difference" means one is subtracted from the other. For example, because 8 = 2^3, x^3 +8 is a sum of cubes.

4. Use the correct formula to factor the sum or difference of cubes. For a difference of cubes, the rule is:

   A^3 - B^3 = (A - B)(A^2 +AB +B^2).

   For a sum of cubes, the rule is:

   A^3 + B^3 = (A + B)(A^2 - AB + B^2).

   So the sum of cubes function x^3 + 8 would factor out as:

   x^3 + 8 =

   x^3 + 2^3 =

   (x + 2)(x^2 - 2x + 4)

5. Factor out the remaining quadratic equation. At this point, you should have either completely factored out your cube function or simplified it to a quadratic function. You can now factor it as you would a normal quadratic. For example, starting with 4x^3 -x, we get:

   4x^3 - x =

   x(4x^2 - 1)

   We can now factor it as a normal difference of squares:

   x(4x^2 - 1) =

   x(2x + 1)(2x -1)

Tips & Warnings

• If you have time, check your math. When you are done factoring, multiply the factors together to see if you end up with the number you started with.

• When using the sum or difference of cubes formula, be careful doing your math. It is easy to accidentally use the total rather than the number being cubed. For example, you might accidentally factor x^3 + 8 as (x + 8)(x^2 - 8x + 64) instead of (x + 2)(x^2 - 2x + 4).