Monday Math 96

A useful polynomial factoring rule is that for positive integer n, the linear term xy 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 xy 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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One Response to “Monday Math 96”

  1. Monday Math 97 « Twisted One 151’s Weblog Says:

    […] 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. […]

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