Shew that $(x + y)^{n} - x^{n} - y^{n}$ is divisible by $x y (x^{2} + x y + y^{2})$ , if $n$ is an odd positive integer not a multiple of $3$ .

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Solution

The expression ${(x + y)}^{n} - x^{n} - y^{n}$ vanishes when $x = 0, y = 0$
Hence it is divisible by $x y$
$x = w y$ , the expression $= y^{n} {{(1 + w)}^{n} - 1 - w^{n}}$
$= - y^{n} {1 + w^{n} + {(w^{2})}^{n}}$
n being odd, the expression vanishes if n is not a multiple of $3$
$x = w^{2} y$ the expression $= - y^{n} {1 + w^{n} + {(w^{2})}^{n}}$
$n \neq 3 k x = w^{2} y$ is a factor
$∴ x y (x - w y) (x - w^{2} y)$
$\Rightarrow x y (x^{2} + y^{2} + x y)$

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