Factorize $27 x^{3} - y^{3} - z^{3} - 9 x y z$

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Solution

$27 x^{3} - y^{3} - z^{3} - 9 x y z = (3 x)^{3} + (- y)^{3} + (- z)^{3} \times 3 x \times (- y) (- z) = (3 x - y - z) [(3 x)^{2} + (- y)^{2} + (- z)^{2} - 3 x \times (- y) - (- y) (- z) - (- z \times 3 x)] = (3 x - y - z) (9 x^{2} + y^{2} + z^{2} + 3 x y - y z + 3 z x)$

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