Factorize: xy^9 - yx^9 - Vidyadip

Question

Factorize: $xy^9 - yx^9$

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Answer

The given expression to be factorized is $xy^9 - yx^9$
This can be wriiten in the form $xy^9 - yx^9 = x.y.y^8 - y.x.x^8$
Take common xy from the two terms of the above expression $xy^9 - yx^9 = xy(y^8 - x^8) = xy(y^8 - x^8)$
$= {xy(y^4)^2 - (x^4)^2)} = xy(y^4 + x^4)(y^4 - x^4) xy^9 - yx^9$
$= xy(y^4 + x^4){(y^2)^2 - (x^2)^2} = xy(y^4 + x^4)(y^2 + x^2)(y^2 - x^2)$
$= xy(y^4 + x^4)(y^2 + x^2){(y)^2 - (x)^2}$
$= xy(y^4 + x^4)(y^2 + x^2)(y + x)(y - x)$
We cannot further factorize the expression.
So, the required factorization of $xy^9 - yx^9 is xy(y^4 + x^4)(y^2 + x^2)(y + x)(y - x)$

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