Common factorizations

These are common factorizations that are used all the time. These should be memorized, but one should also know how they are derived.

Basic Factorizations

$x^2-y^2=(x+y)(x-y)$

$x^3+y^3=(x+y)(x^2-xy+y^2)$

$x^3-y^3=(x-y)(x^2+xy+y^2)$

Vieta's/Newton Factorizations

These factorizations are useful for problem that could otherwise be solved by Newton sums or problems that give a polynomial, and ask a question about the roots. Combined with Vieta's formulas, these are excellent factorizations that show up everywhere.