sum of cubes

Unlike a sum of squares, a sum of two cubes always factors over the real numbers, and it has its own fixed formula worth memorizing. Whenever you see one perfect cube added to another, a^3 + b^3, it splits into a binomial times a trinomial: a^3 + b^3 = (a + b)(a^2 - ab + b^2).

The structure is easy to remember with a rhythm. The binomial is the sum of the cube roots, (a + b). The trinomial is built from those roots as: first squared, minus their product, plus second squared. A handy mnemonic is 'SOAP': Same sign as the original (here +), then Opposite sign, then Always Positive — giving (a + b)(a^2 - ab + b^2).

Two cautions. First, the trinomial a^2 - ab + b^2 generally does NOT factor further over the reals, so do not waste effort trying. Second, make sure both terms really are perfect cubes (like 8 = 2^3 or 27x^3 = (3x)^3) before applying the formula; otherwise it does not apply.