Factorise $27 a^{3} + 8 b^{3} - 18 a^{2} b - 12 a b^{2}$

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Solution

Consider, $27 a^{3} + 8 b^{3} - 18 a^{2} b - 12 a b^{2}$
$= 27 a^{3} - 18 a^{2} b - 12 a b^{2} + 8 b^{3}$
$= 9 a^{2} (3 a - 2 b) - 4 b^{2} (3 a - 2 b)$
$= (9 a^{2} - 4 b^{2}) (3 a - 2 b)$
$= ((3 a)^{2} - (2 b)^{2}) (3 a - 2 b)$
$= (3 a + 2 b) (3 a - 2 b) (3 a - 2 b)$ ------ since, $a^{2} - b^{2} = (a + b) (a - b)$
$= (3 a - 2 b)^{2} (3 a + 2 b)$

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