Factorize the following expression:

$x y^{9} - y x^{9}$

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Solution

$x y^{9} - y x^{9} = x y (y^{8} - x^{8}) = x y [(y^{4})^{2} - (x^{4})^{2}] {∵ a^{2} - b^{2} = (a + b) (a - b)} = x y (y^{4} + x^{4}) (y^{4} - x^{4}) = x y (y^{4} + x^{4}) {(y^{2})^{2} - (x^{2})^{2}} = x y (y^{4} + x^{4}) (y^{2} + x^{2}) (y^{2} - x^{2}) = x y (y^{4} + x^{4}) (y^{2} + x^{2}) (y + x) (y - x)$

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