Factorise: $a^{3} - b^{3} - c^{3} - 3 a b c$ .

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Solution

Factorise the polynomial
Using the identity $x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - x z)$
put $x = a, y = - b, z = - c$
$a^{3} - b^{3} - c^{3} - 3 a b c = (a - b - c) (a^{2} + {(- b)}^{2} + {(- c)}^{2} - a (- b) - (- b) (- c) - (- c) a) \Rightarrow a^{3} - b^{3} - c^{3} - 3 a b c = (a - b - c) (a^{2} + b^{2} + c^{2} + a b - b c + a c)$
Hence, the factors of $a^{3} - b^{3} - c^{3} - 3 a b c$ are $(a - b - c)$ and $(a^{2} + b^{2} + c^{2} + a b - b c + a c)$ .

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