Find the following product :
$(2 x - y + 3 z) (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z)$

8 x^{3} - y^{3} + 27 z^{3} + 18 x y z

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8 x^{3} + y^{3} - 27 z^{3} - 18 x y z

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- 8 x^{3} + y^{3} + z^{3} - 18 x y z

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8 x^{3} - y^{3} + 27 z^{3} - 18 x y z

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Solution

The correct option is A $8 x^{3} - y^{3} + 27 z^{3} + 18 x y z$
We know that,
$a^{3} + b^{3} + c^{3} = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) + 3 a b c$
Thus,
$(2 x - y + 3 z) (4 x^{2} + y^{2} + 9 z^{2} + 2 x y + 3 y z - 6 x z)$
Here, $a = 2 x$ , $b = - y$ and $c = 3 z$
$= (2 x)^{3} - y^{3} + (3 z)^{3} - 3 (2 x) (- y) (3 z)$
$= 8 x^{3} - y^{3} + 27 z^{3} + 18 x y z$

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