State true or false:
$x^{3} + 27 y^{3} + 8 z^{3} - 18 x y z$ $= (x + 3 y + 2 z) (x^{2} + 9 y^{2} + 4 z^{2} - 3 x y z - 6 y z - 2 z x)$

True

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False

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Solution

The correct option is A True
$(x + 3 y + 2 z) (x^{2} + 9 y^{2} + 4 z^{2} - 3 x y - 6 y z - 2 z x)$

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

$= x^{3} + 9 x y^{2} + 4 x z^{2} - 3 x^{2} y - 6 x y z - 2 z x^{2} + 3 y x^{2} + 27 y^{3} + 12 y z^{2} - 9 x y^{2} - 18 y^{2} z - 6 z x y + 2 z x^{2} + 18 z y^{2} + 8 z^{3}$
$- 6 x y z - 12 y z^{2} - 4 z^{2} x$

$= (x^{3} + 27 y^{3} + 8 z^{3}) + (9 x y^{2} - 9 x y^{2}) + (4 x z^{2} - 4 z^{2} x) + (- 3 x^{2} y + 3 y x^{2}) + (- 6 x y z - 6 z x y - 6 x y z)$
$+ (- 2 z x^{2} + 2 z x^{2}) + (12 y z^{2} - 12 y z^{2}) + (- 18 y^{2} z + 18 z y^{2})$

$= (x^{3} + 27 y^{3} + 8 z^{3}) + 0 + 0 + 0 - 18 x y z + 0 + 0 + 0 = x^{3} + 27 y^{3} + 8 z^{3} - 18 x y z$

Hence, $x^{3} + 27 y^{3} + 8 z^{3} - 18 x y z = (x + 3 y + 2 z) (x^{2} + 9 y^{2} + 4 z^{2} - 3 x y - 6 y z - 2 z x)$

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