Find the factors of the polynomial $8 a^{3} + 27 b^{3} + 64 c^{3} - 72 a b c$ .

(2 a + 3 b + 4 c) (4 a^{2} + 9 b^{2} + 16 c^{2} - 6 a b + 12 b c - 8 a c)

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(2 a + 3 b + 4 c) (4 a^{2} + 9 b^{2} + 16 c^{2} + 6 a b - 12 b c - 8 a c)

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(2 a + 3 b + 4 c) (4 a^{2} + 9 b^{2} + 16 c^{2} - 6 a b - 12 b c - 8 a c)

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(2 a + 3 b + 4 c) (4 a^{2} + 9 b^{2} + 16 c^{2} - 6 a b - 12 b c + 8 a c)

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Solution

The correct option is B $(2 a + 3 b + 4 c) (4 a^{2} + 9 b^{2} + 16 c^{2} - 6 a b - 12 b c - 8 a c)$
We have to factorize $8 a^{3} + 27 b^{3} + 64 c^{3} - 72 a b c$
$\Rightarrow 8 a^{3} + 27 b^{3} + 64 c^{3} - 72 a b c = (2 a)^{3} + (3 b)^{3} + (4 c)^{3} - 3 (2 a) (3 b) (4 c)$
We know that $x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)$
So we can write given expression as $(2 a + 3 b + 4 c) (4 a^{2} + 9 b^{2} + 16 c^{2} - 6 a b - 12 b c - 8 a c)$
Therefore option $C$ is correct

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