# Sum and difference of powers

The **sum and difference of powers** are powerful factoring techniques that, respectively, factor a sum or a difference of certain powers.

## Contents

## Sums of Odd Powers

## Differences of Powers

If is a positive integer and and are real numbers,

For example:

Note that the number of terms in the second factor is equal to the exponent in the expression being factored.

An amazing thing happens when and differ by , say, . Then and

.

For example:

If we also know that then:

## Sum of Cubes

## Factorizations of Sums of Powers

Note that all these sums of powers can be factorized as follows:

If we have a difference of powers of degree , then

Note, of course, that some of the signs simply change when we have sum of powers instead of difference. Regardless, observe that the "longer" polynomial in the factorization is simply a binomial theorem expansion of the binomial , except for the fact that the coefficient on each of the terms is . This can be quite useful in problems that might have a sum of powers expression as well as an application of the binomial theorem.

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## See Also

- Factoring
- Difference of squares, an extremely common specific case of this.
- Binomial Theorem