Find the HCF of the polynomials $x^{4} - y^{4} a n d x^{2} (x - y) + y^{2} (x - y) + y - x .$

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Solution

Factors of $x^{4} - y^{4} = (x^{2} - y^{2}) (x^{2} + y^{2})$
$= (x - y) (x + y) (x^{2} + y^{2})$

Factors of $x^{2} (x - y) + y^{2} (x - y) + y - x$
$= x^{2} (x - y) + y^{2} (x - y) - (x - y)$
$= (x - y) (x^{2} + y^{2} - 1)$

Common factors of $(x^{4} - y^{4}, x^{2} (x - y) + y^{2} (x - y) + y - x) = x - y$

$∴$ H.C.F of $(x^{4} - y^{4}, x^{2} (x - y) + y^{2} (x - y) + y - x) = x - y$

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