Factoring cubic equations is significantly more challenging than factoring quadratics – there are no guaranteed-to-work methods like guess-and-check and the box method, and the cubic equation, unlike the quadratic equation, is so lengthy and convoluted that it is almost never taught in math classes. Fortunately, there are simple formulas for two types of cubics: the sum of cubes and the difference of cubes. These binomials always factor into the product of a binomial and a trinomial.
Sum of Cubes
Step 1
Take the cube root of the two binomial terms. The cube root of A is the number that, when cubed, is equal to A; for example, the cube root of 27 is 3 because 3 cubed is 27. The cube root of x^3 is simply x.
Step 2
Write the sum of the cube roots of the two terms as the first factor. For example, in the sum of cubes “x^3 + 27,” the two cube roots are x and 3, respectively. The first factor is therefore (x + 3).
Step 3
Square the two cube roots to get the first and third term of the second factor. Multiply the two cube roots together to get the second term of the second factor. In the above example, the first and third terms are x^2 and 9, respectively (3 squared is 9). The middle term is 3x.
Step 4
Write out the second factor as the first term minus the second term plus the third term. In the above example, the second factor is (x^2 – 3x + 9). Multiply the two factors together to get the factored form of the binomial: (x + 3)(x^2 – 3x + 9) in the example equation.
Difference of Cubes
Step 1
Take the cube root of the two binomial terms. The cube root of A is the number that, when cubed, is equal to A; for example, the cube root of 27 is 3 because 3 cubed is 27. The cube root of x^3 is simply x.
Step 2
Write the difference of the cube roots of the two terms as the first factor. For example, in the difference of cubes “8x^3 – 8,” the two cube roots are 2x and 2, respectively. The first factor is therefore (2x – 2).
Step 3
Square the two cube roots to get the first and third term of the second factor. Multiply the two cube roots together to get the second term of the second factor. In the above example, the first and third terms are 4x^2 and 4, respectively (2 squared is 4). The middle term is 4x.
Step 4
Write out the second factor as the first term minus the second term plus the third term. In the above example, the second factor is (x^2 + 4x + 4). Multiply the two factors together to get the factored form of the binomial: (2x – 2)(4x^2 + 4x + 4) in the example equation.
