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.
Differences and Sums of Powers
Using the formula for the sum of a geometric sequence, it's easy to derive the more general formula:
Take note of the specific case where n is odd:
This also leads to the formula for the sum of cubes,
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.
Other Useful Factorizations
- See Simon's Favorite Factoring Trick (This is not a recognized formula, please do not quote it on contests)
- Binomial theorem
- Prove that is never divisible by 121 for any positive integer
- Prove that is divisible by 7 - USSR Problem Book