Find the product:
$(x + 2 y + 3 z) (x^{2} + 4 y^{2} + 9 z^{2} - 2 x y - 6 y z - 3 z x)$

x^{3} + 8 y^{3} + 27 z^{3} + 18 x y z

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x^{3} + 8 y^{3} + 27 z^{3} - 18 x y z

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x^{3} + 8 y^{3} - 27 z^{3} - 18 x y z

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x^{3} - 8 y^{3} + 27 z^{3} - 18 x y z

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Solution

The correct option is B $x^{3} + 8 y^{3} + 27 z^{3} - 18 x y z$
Identity used: $x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)$
Now, $(x + 2 y + 3 z) (x^{2} + 4 y^{2} + 9 z^{2} - 2 x y - 6 y z - 3 z x)$
$= (x + 2 y + 3 z) (x^{2} + (2 y)^{2} + (3 z)^{2} - x \times 2 y - 2 y \times 3 z - 3 z \times x)$
$= x^{3} + (2 y)^{3} + (3 z)^{2} - 3 \times x \times 2 y \times 3 z$
$= x^{3} + 8 y^{3} + 27 z^{3} - 18 x y z$

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