Differences and Sums of Powers

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The formulas for the differences and sums of powers for factoring polynomials.

Author: Jaden Park

Formulas

The general formula for the difference of powers is:

a^{n} - b^{n} = (a - b)(a^{n - 1} + a^{n - 2} b + \ldots + ab^{n - 2} + b^{n - 1})

The general formula for the sum of powers (only if $n$ is odd) is:

a^{n} + b^{n} = (a + b)(a^{n - 1} - a^{n - 2} b + a^{n - 3} b^{2} - a^{n - 4} b^{3} + \ldots + b^{n - 1}).

Note: There is no formula for the sum of powers if $n$ is even.

Pretty common, especially for grade 9 - 11 contests, so I would memorize up to $n = 3.$

Binomial Theorem

SFFT