# Factoring

**Factoring** is an essential part of any mathematical toolbox. To factor, or to break an expression into factors, is to write the expression (often an integer or polynomial) as a product of different terms. This often allows one to find information about an expression that was not otherwise obvious.

## Contents

## Differences and Sums of Powers

Using the formula for the sum of a geometric sequence, it's easy to derive the general formula for difference of powers:

If , this creates the difference of squares factorization,

This leads to the difference of cubes factorization,

In addition, if is odd:

This also leads to the formula for the sum of cubes,

Another way to discover these factorizations is the following: the expression is equal to zero if . If one factorizes a product which is equal to zero, one of the factors must be equal to zero, so must have a factor of . Similarly, we note that the expression when is odd is equal to zero if , so it must have a factor of . Note that when is even, , rather than 0, so this gives us no useful information.

## 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

## Practice Problems

- Prove that is never divisible by 121 for any positive integer .
- Prove that is divisible by 7. - USSR Problem Book
- Factor .
- Factor into two polynomials with real coefficients.
- Given that , prove that .