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).