Question
Factorise$ : x^4+y^4-27 x^2 y^2$
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Answer
$ x^4+y^4-27 x^2 y^2$
$ =\left(x^2\right)^2+\left(y^2\right)^2-2 x^2 y^2-25 x^2 y^2$
$ =\left(x^2-y^2\right)^2-25 x^2 y^2$
$ =\left(x^2-y^2\right)^2-(5 x y)^2 \quad[\because \mathrm{a} 2-\mathrm{b} 2=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})]$
$ =\left[\left(x^2-y^2\right)+5 x y\right]\left[\left(x^2-y^2\right)-5 x y\right]$
$ =\left[x^2+5 x y-y^2\right]\left[x^2-5 x y-y^2\right]$
$ =\left(x^2\right)^2+\left(y^2\right)^2-2 x^2 y^2-25 x^2 y^2$
$ =\left(x^2-y^2\right)^2-25 x^2 y^2$
$ =\left(x^2-y^2\right)^2-(5 x y)^2 \quad[\because \mathrm{a} 2-\mathrm{b} 2=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})]$
$ =\left[\left(x^2-y^2\right)+5 x y\right]\left[\left(x^2-y^2\right)-5 x y\right]$
$ =\left[x^2+5 x y-y^2\right]\left[x^2-5 x y-y^2\right]$
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