A useful polynomial factoring rule is that for positive integer *n*, the linear term *x*–*y* is a factor of . More specifically,

;

expanding the right hand side of the above gives you the left after cancellation of various terms.

For example, for the first few values of *n*

;

;

;

;

;

and so on.

It is this which gives us the formula for a finite geometric series; letting *x*=*r*, *y*=1, and our power be *n*+1, then

.

Further, note that if *x* and *y* are integers, then is also an integer, and thus the integer *x*–*y* is a factor of the integer .

Now, let *n* be odd. If we let *x*=*a* and *y*=-*b*, then

becomes

For example,

;

;

;

and so on. And we see that if *x* and *y* are integers, then is also an integer, and thus the integer *x*+*y* is a factor of the integer when *n* is odd.

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Tags: Factoring, Geometric Series, Math, Monday Math

This entry was posted on November 9, 2009 at 1:09 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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November 16, 2009 at 1:38 am |

[…] odd factor greater than 1. So let n=ab, with b>1 odd; we see 1≤a<n and 1<a≤n. Now, recall that for integers x and y and odd positive integer m, that is divisible by the integer x+y. […]