## 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.