Factorise: $8 p^{3} - q^{3} - 12 p^{2} q + 6 p q^{2}$ .

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Solution

We are given,
$8 p^{3} - q^{3} - 12 p^{2} q + 6 p q^{2}$
$= (2 p)^{3} - (q)^{3} - 6 p q (2 p - q)$
$= (2 p)^{3} - (q)^{3} - 3 (2 p) (q) (2 p - q)$
Let $a = 2 p =$ & $b = q$
So,
$(a)^{3} - (b)^{3} - 3 a b (a - b)$
$= (a - b)^{3}$
So,
$(2 p)^{3} - (q)^{3} - 3 (2 p) (q) (2 p - q) = (2 p - q)^{3}$
So,
$8 p^{3} - q^{3} - 12 p^{2} q + 6 p a^{2}$
$= (2 p - q) (2 p - q) (2 p - q)$

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