# Difference between revisions of "Sum and difference of powers"

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

− | If we have a sum of powers of degree | + | If we have a sum of powers of degree <math>n</math>, then |

− | + | <cmath>x^n±y^n=(x±y)(x^n+x^n-1y+x^n-2y^2...+y^n)</cmath> | |

− | Note that the "longer" polynomial in the factorization is simply a binomial theorem expansion of the binomial (x+y)^n, except for the fact that the coefficient on each of the terms is 1. This can be quite useful in problems that might have a sum of powers expression as well as an application of the binomial theorem. | + | Note that the "longer" polynomial in the factorization is simply a binomial theorem expansion of the binomial <math>(x+y)^n</math>, except for the fact that the coefficient on each of the terms is <math>1</math>. This can be quite useful in problems that might have a sum of powers expression as well as an application of the binomial theorem. |

- icecreamrolls8 | - icecreamrolls8 |

## Revision as of 14:54, 5 May 2021

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 *long* 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 sum of powers of degree , then

Note 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.

- icecreamrolls8

## See Also

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

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