How to Factor Cubes

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

Factoring cubes can be done in three basic steps. First, factor out any common terms in the cubic function. Then, look for a difference or sum of cubes and factor it. At this point, if the problem is not factored out as far as it can be, the complex term will be a quadratic equation. You can then solve that term in the way you would solve a quadratic equation. Use the techniques you already use for solving quadratic equations to simplify the equation further.

Instructions

1. Learn how to factor a quadratic equation. In many cases, you need to solve a quadratic equation as one step in solving a cubic equation (see Resources).

2. Factor out any common terms. A common term is a number or variable that is a component of every term in the problem. For example, in the expression x^3 +2x^2 +x, x is a common term that can be factored out:

   x^3 +2x^2 + x = x ( x^2 + x +1).

   In the expression 2x^3 + 54, 2 can be factored out:

   2x^3 + 54 = 2(x^3 + 27).

3. Express the problem as a sum of cubes or as a difference of cubes, if possible. A sum of cubes is two cubed numbers added to each other. For example, the problem x^3 + 27 = x^3 + 3^3, since 3^3 = 27. A difference of cube is one cubed number subtracted from another, such as x^3 - 27 = x^3 - 3^3.

4. Apply the rule for a difference of cubes or a sum of cubes. The difference of cubes rule is A^3 - B^3 = (A - B)(A^2 +AB +B^2). The sum of cubes rules is A^3 + B^3 = (A + B)(A^2 - AB + B^2). For example, if the expression is x^3 - 3^3, x is the A term and 3 is the B term. You get:

   x^3 - 27 =

   x^3 - 3^3 =

   (x - 3)(x^2 +3x + 3^2) =

   (x-3)(x^2 + 3x + 9)

5. Look for differences of squares or other quadratic patterns to factor out further terms from the equation. For example, you can factor an x out of x^3 - 16x, resulting in:

   x^3 - 16x =

   x(X^2 - 16)

   Since x^2 - 16 is a difference of squares, we get:

   x(x^2 - 16) =

   x(x + 4)(X - 4)

Tips & Warnings

• Just because you have a complex term doesn't mean you can factor it. For example, 3x^3 + 2x^2 + 84 can't be factored.

Resources

• Read this Article in Spanish
• Quartic/Cubic/Quadratic Equation Calculator