Find the product using appropriate identity:
$(x - 2 y - z) (x^{2} + 4 y^{2} + z^{2} + 2 x y - 2 y z + z x)$

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Solution

We know the identity $(a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) = a^{3} + b^{3} + c^{3} - 3 a b c$
Using the above identity and taking $a = x$ , $b = - 2 y$ and $c = - z$ , the product $(x - 2 y - z) (x^{2} + 4 y^{2} + z^{2} + 2 x y - 2 y z + z x)$ can be computed as follows:
$(x - 2 y - z) (x^{2} + 4 y^{2} + z^{2} + 2 x y - 2 y z + z x) = x^{3} + (- 2 y)^{3} + (- z)^{3} - (3 x \times - 2 y \times - z) = x^{3} - 8 y^{3} - z^{3} - 6 x y z$
Hence, $(x - 2 y - z) (x^{2} + 4 y^{2} + z^{2} + 2 x y - 2 y z + z x) = x^{3} - 8 y^{3} - z^{3} - 6 x y z$

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