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

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==Differences of Powers== | ==Differences of Powers== | ||

− | If p is a positive integer and x and y are real numbers, | + | If <math>p</math> is a positive integer and <math>x</math> and <math>y</math> are real numbers, |

<math>x^{p+1}-y^{p+1}=(x-y)(x^p+x^{p-1}y+\cdots +xy^{p-1}+y^p)</math> | <math>x^{p+1}-y^{p+1}=(x-y)(x^p+x^{p-1}y+\cdots +xy^{p-1}+y^p)</math> | ||

− | For example | + | For example: |

<math>x^2-y^2=(x-y)(x+y)</math> | <math>x^2-y^2=(x-y)(x+y)</math> | ||

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Note that the number of terms in the ''long'' factor is equal to the exponent in the expression being factored. | 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 x and y differ by 1, say, x = y+1. Then x-y = 1 and | + | An amazing thing happens when <math>x</math> and <math>y</math> differ by <math>1</math>, say, <math>x = y+1</math>. Then <math>x-y = 1</math> and |

<math>x^{p+1}-y^{p+1}=(y+1)^{p+1}-y^{p+1}</math> | <math>x^{p+1}-y^{p+1}=(y+1)^{p+1}-y^{p+1}</math> | ||

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<math>=(y+1)^p+(y+1)^{p-1}y+\cdots +(y+1)y^{p-1} +y^p</math>. | <math>=(y+1)^p+(y+1)^{p-1}y+\cdots +(y+1)y^{p-1} +y^p</math>. | ||

− | For example | + | For example: |

<math>(y+1)^2-y^2=(y+1)+y</math> | <math>(y+1)^2-y^2=(y+1)+y</math> | ||

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<math>(y+1)^4-y^4=(y+1)^3+(y+1)^2y+(y+1)y^2+y^3</math> | <math>(y+1)^4-y^4=(y+1)^3+(y+1)^2y+(y+1)y^2+y^3</math> | ||

− | If we also know that <math>y\geq 0</math> then | + | If we also know that <math>y\geq 0</math> then: |

<math>2y\leq (y+1)^2-y^2\leq 2(y+1)</math> | <math>2y\leq (y+1)^2-y^2\leq 2(y+1)</math> |

## Revision as of 18:43, 8 July 2008

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

## Sums of 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:

## See Also

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

*This article is a stub. Help us out by expanding it.*