Differences of powers

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If p is a positive integer and x and y are real numbers,

Img2.gif

For example,

Img3.gif

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

Img4.gif

For example,

Img5.gif

If we also know that Img6.gif then

Img7.gif