Factorize each of the following expressions:$2 \sqrt2 a^3+ 16\sqrt2 b^3 + c^3 - 12abc$

Given:

$2 \sqrt2 a^3+ 16\sqrt2 b^3 + c^3 - 12abc$

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,

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

$=(\sqrt{2} a+2 \sqrt{2} b+c)(2 a^{2}+8 b^{2}+c^{2}-4 a b-2 \sqrt{2} b c-\sqrt{2} c a)$

Hence, $(2 \sqrt{2} a)^{3}+(2 \sqrt{2} b)^{3}+(c)^{3}-3 \times \sqrt{2} a \times 2 \sqrt{2} b \times c = (\sqrt{2} a+2 \sqrt{2} b+c)(2 a^{2}+8 b^{2}+c^{2}-4 a b-2 \sqrt{2} b c-\sqrt{2} c a)$.

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Updated on: 10-Oct-2022

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