Factorise:
$8 (x + y)^{3} - 27 (x - y)^{3}$

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Solution

$8 (x + y)^{3} - 27 (x - y)^{3}$
$= (2 x + y)^{3} + (3 (x - y))^{3} [∵ (a + b)^{2} = a^{2} + b^{2} + 2 a b, (a - b)^{2} = a^{2} + b^{2} - 2 a b]$
$= 2 (x + y) + 3 (x - y) [4 (x + y)^{2} - 6 (x + y) (x - y) + 9 (x - y)^{2}$ $[∵ a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2}]$
$= (2 x + 2 y + (3 x - 3 y)) ((2 x + 2 y)^{2} + (3 x - 3 y)^{2} - (2 x + 2 y) (3 x - 3 y))$

$= (5 x - y) (7 x^{2} + 19 y^{2} - 10 x y)$

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