Factorize each of the following expressions:$ 3 \sqrt{3} a^{3}-b^{3}-5 \sqrt{5} c^{3}-3 \sqrt{15} a b c $

Given:

\( 3 \sqrt{3} a^{3}-b^{3}-5 \sqrt{5} c^{3}-3 \sqrt{15} a b c \)

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 $a + b + c = 0$

Therefore,

$3 \sqrt{3} a^{3}-b^{3}-5 \sqrt{5} c^{3}-3 \sqrt{15} a b c = (\sqrt{3} a)^{3}+(-b)^{3}+(-\sqrt{5} c)^{3}-3 \sqrt{3} a \times(-b) \times(-\sqrt{5} c)$

$=(\sqrt{3} a-b-\sqrt{5} c)[(\sqrt{3} a)^{2}+(-b)^{2}+(-\sqrt{5} c)^{2}-\sqrt{3} a \times(-b)-(-b)(-\sqrt{5} c)-(-\sqrt{5} c) \times \sqrt{3} a$

$=(\sqrt{3} a-b-\sqrt{5} c)(3 a^{2}+b^{2}+5 c^{2}+\sqrt{3} a b-\sqrt{5} b c+\sqrt{15} c a)$

Hence, $3 \sqrt{3} a^{3}-b^{3}-5 \sqrt{5} c^{3}-3 \sqrt{15} a b c = (\sqrt{3} a-b-\sqrt{5} c)(3 a^{2}+b^{2}+5 c^{2}+\sqrt{3} a b-\sqrt{5} b c+\sqrt{15} c a)$.