Factoring xⁿ ± yⁿ — Cheat sheet

Key method

For integer $n\ge1$,

For odd $n$, the corresponding sum factors with alternating signs in the second factor.

Example

For $n=5$,

On multiplication, every mixed term cancels. For $x^6-y^6$, first use the difference of squares $(x^3-y^3)(x^3+y^3)$, then apply both cube identities.

Avoid this mistake

The same sum formula does not work for even exponents: substituting $x=-y$ into $x^2+y^2$ usually does not give zero.