Factorize each of the following expressions:$(a - 3b)^3 + (3b - c)^3 + (c - a)^3$

Given:

$(a - 3b)^3 + (3b - c)^3 + (c - a)^3$

To do:

We have to multiply the given expressions.

Solution:

We know that,

$a^3 + b^3 + c^3 - 3abc = (a + b + c) (a^2 + b^2 + c^2 - ab - bc - ca)$

$a^3 + b^3 + c^3 = 3abc$ if $a + b + c = 0$

Here,

$a - 3b + 3b - c + c - a = 0$

Therefore,

$(a - 3b)^3 + (3b - c)^3 + (c - a)^3 = 3 (a - 3b) (3b - c) (c - a)$

Hence, $(a - 3b)^3 + (3b - c)^3 + (c - a)^3 = 3 (a - 3b) (3b - c) (c - a)$.