An Overview of the Methods — Cheat sheet

Key method

A practical order is: common factor, familiar identity, grouping or trinomial, then substitution. Verify every proposed factorization by multiplication. Over the reals, some polynomials stop at irreducible quadratic factors.

Example

Factor $2x^3-8x$. Every term contains $2x$, giving $2x(x^2-4)$. The remaining expression is a difference of squares, so

To solve $2x^3-8x=0$, set each factor equal to zero: $x=0,2,-2$. Expanding the answer recovers both original terms.

Avoid this mistake

Factoring is an identity, not division. Dividing an equation by $x$ without checking $x=0$ can lose a solution.