# Factoring

Note to readers and editers: Please fix up this page by adding in material from Joe's awesome factoring page.

## Contents

### Why Factor

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,

## Vieta's/Newton Factorizations

These factorizations are useful for problems 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- See Simon's Favorite Factoring Trick (This is not a recognized formula, please do not quote it on contests)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

## Practice Problems

- Prove that is never divisible by 121 for any positive integer .
- Prove that is divisible by 7. - USSR Problem Book
- Factor .