Factorise each of the following
$8 a^{3} - b^{3} - 12 a^{2} b^{2} + 6 a b^{2}$

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Solution

$8 a^{3} - b^{3} - 12 a^{2} b + 6 a b^{2}$
= $(2 a)^{3} + (- b)^{3} + 3 (2 a)^{2} (- b) + 2 (2 a) (- b)^{2}$
Suitable identities is $x^{3} - y^{3} - 3 x^{2} y + 3 x y^{2} = (x - y)^{3}$
$∴ (2 a)^{3} + (- b)^{3} + 3 (2 a)^{2} (- b) + 2 (2 a) (- b)^{2}$
= $(2 a - b)^{3}$

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