Simplify $(x + 2 y + 3 z) (x^{2} + 4 y^{2} + 9 z^{2} - 2 x y - 6 y z - 3 z x)$

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Solution

We know that, $(x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x) = x^{3} + y^{3} + z^{3} - 3 x y z$
$∴ (x + 2 y + 3 z) (x^{2} + 4 y^{2} + 9 z^{2} - 2 x y - 6 y z - 3 z x)$
$= (x + 2 y + 3 z) [x^{2} + (2 y)^{2} + (3 z)^{2} - (x) (2 y) - (2 y) (3 z) - (3 z) (x)]$
$= (x^{3}) + (2 y)^{3} + (3 z)^{3} - 3 (x) (2 y) (3 z)$
$= x^{3} + 8 y^{3} + 27 z^{3} - 18 x y z$

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