Factorise the following: $- 27 x^{3} + y^{3} - z^{3} - 9 x y z$

(- 3 x + y - z) (9 x^{2} - y^{2} - z^{2} + 3 x y + y z - 3 x z)

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(3 x + y - z) (9 x^{2} + y^{2} + z^{2} + x y + y z - 3 x z)

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(- 3 x + y - z) (9 x^{2} + y^{2} + z^{2} + 3 x y + y z - 3 x z)

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(- y - 3 z) (9 x^{2} + y + z^{2} + 3 x y + y - 3 z)

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Solution

The correct option is C $(- 3 x + y - z) (9 x^{2} + y^{2} + z^{2} + 3 x y + y z - 3 x z)$
$- 27 x^{3} + y^{3} - z^{3} - 9 x y z$
Using, $a^{3} + b^{3} + c^{3} - 3 a b c = (a + b + c) (a^{2} + b^{2} + c^{2} - a b - a c - b c)$
$= (- 3 x)^{3} + y^{3} + (- z^{3}) - 3 (- 3 x) (y) (- z)$
$= (- 3 x + y - z) ((- 3 x)^{2} + y^{2} + (- z)^{2} - (- 3 x) y - (- 3 x) (- z) - y (- z))$
$= (- 3 x + y - z) (9 x^{2} + y^{2} + z^{2} + 3 x y - 3 x z + y z)$

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