Find the factors of $x^{3} - 8 y^{3} - 27 - 18 x y$ .

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Solution

We write this as
$x^{3} + {(- 2 y)}^{3} + {(- 3)}^{3} - 3 x (- 2 y) (- 3)$
This is in the form $a^{3} + b^{3} + c^{3} - 3 a b c$ , where $a = x . b = - 2 y$ and $c = - 3$ . Hence
$x^{3} - 8 y^{3} - 27 - 18 x y = (x - 2 y - 3) (x^{2} + 4 y^{2} + 9 + 2 x y - 6 y + 3 x)$

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