Difference of Two Powers/Examples/Difference of Two Cubes

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Theorem

$x^3 - y^3 = \paren {x - y} \paren {x^2 + x y + y^2}$


Corollary

$x^3 - 1 = \paren {x - 1} \paren {x^2 + x + 1}$


Proof

From Difference of Two Powers:

$\displaystyle a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$

The result follows directly by setting $n = 3$.

$\blacksquare$


Sources