Note to readers and editers: Please fix up this page by adding in material from Joe's awesome factoring page.
Factoring equations is an essential part of problem solving. Applying number theory to products yields many results.
There are many ways to factor.
Difference of Squares
Difference of Cubes
Sum of Cubes
See Simon's Favorite Factoring Trick (This is not a recognized formula, please do not quote it on the USAMO or similar national proof contests)
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.