Sum of Two Odd Powers/Examples/Sum of Two Cubes/Proof 2
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Theorem
- $x^3 + y^3 = \paren {x + y} \paren {x^2 - x y + y^2}$
Proof
From Sum of Two Odd Powers:
- $a^{2 n + 1} + b^{2 n + 1} = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb + a b^{2 n - 1} + b^{2 n} }$
We have that $3 = 2 \times 1 + 1$.
Hence setting $n = 1$ gives the result.
$\blacksquare$