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

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

− | == | + | ==Sums of Powers== |

− | + | <math>a^{2n+1}+b^{2n+1}=(a+b)(a^{2n}-a^{2n-1}b+a^{2n-2}b^2-\ldots-ab^{2n-1}+b^{2n})</math> | |

− | + | ||

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

+ | If p is a positive integer and x and y 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> | ||

+ | |||

+ | For example, | ||

+ | |||

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

+ | |||

+ | <math>x^3-y^3=(x-y)(x^2+xy+y^2)</math> | ||

+ | |||

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

+ | |||

+ | 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 | ||

+ | |||

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

+ | |||

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

+ | |||

+ | For example, | ||

+ | |||

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

+ | |||

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

+ | |||

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

+ | |||

+ | <math>3y^2\leq (y+1)^3-y^3\leq 3(y+1)^2</math> | ||

+ | |||

+ | <math>4y^3\leq (y+1)^4-y^4\leq 4(y+1)^3</math> | ||

+ | |||

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

+ | |||

==See Also== | ==See Also== | ||

* [[Factoring]] | * [[Factoring]] |

## Revision as of 09:04, 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 p is a positive integer and x and y 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 x and y differ by 1, say, x = y+1. Then x-y = 1 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.*