A binomial is any mathematical expression with only two terms, such as “x + 5.” A cubic binomial is a binomial where one or both of the terms is something raised to the third power, such as “x^3 + 5,” or “y^3 + 27.” (Note that 27 is three to the third power, or 3^3.) When the task is to “simplify a cube (or cubic) binomial,” this usually refer to one of three situations: (1) an entire binomial term is cubed, as in “(a + b)^3” or “(a – b)^3”; (2) each of the terms of a binomial is cubed separately, as in “a^3 + b^3” or “a^3 – b^3”; or (3) all other situations in which the highest-power term of a binomial is cubed. There are specialty formulas to handle the first two situations, and a straightforward method to handle the third.

Determine which of the five basic kinds of cubic binomial you are working with: (1) cubing a binomial sum, such as “(a + b)^3”; (2) cubing a binomial difference, such as “(a – b)^3”; (3) the binomial sum of cubes, such as “a^3 + b^3”; (4) the binomial difference of cubes, such as “a^3 – b^3”; or (5) any other binomial where the highest power of either of the two terms is 3.

In cubing a binomial sum, make use of the following equation:

(a + b)^3 = a^3 + 3(a^2)b + 3a(b^2) + b^3.

In cubing a binomial difference, make use of the following equation:

(a - b)^3 = a^3 - 3(a^2)b + 3a(b^2) - b^3.

In working with the binomial sum of cubes, make use of the following equation:

a^3 + b^3 = (a + b) (a^2 – ab + b^2).

In working with the binomial difference of cubes, make use of the following equation:

a^3 - b^3 = (a - b) (a^2 + ab + b^2).

In working with any other cubic binomial, with one exception, the binomial cannot be further simplified. The exception involves situations where both terms of the binomial involve the same variable, such as “x^3 + x,” or “x^3 – x^2.” In such cases, you may factor out the lowest-powered term. For example:

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

x^3 – x^2 = x^2(x – 1).