If $x = p + q, y = p w + q w^{2}$ and $z = p w^{2} + q w$ where w is a complex cube root of unity then xyz=

p^{3} + q^{3}

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p^{2} - p q + q^{2}

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1 + p^{3} + q^{3}

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p^{3} - q^{3}

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Solution

The correct option is A $p^{3} + q^{3}$
$x y z = (p + q) (p w + q w^{2}) (p w^{2} + q w)$
$= (p + q) (p^{2} w^{3} + p q w^{2} + p q w^{4} + q^{2} w^{3})$
Since $w^{3} = 1$ and $w^{2} = w$
$x y z = (p + q) (p^{2} + q^{2} + p q (w^{2} + w^{3} . w))$
$= (p + q) (p^{2} + q^{2} + p q (w^{2} + w))$
$= (p + q) (p^{2} + q^{2} - p q)$ ... since $(w + w^{2} + 1 = 0)$
$= p^{3} + q^{3}$

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