Question
Factorise the following using the identity $a^2- b^2= (a + b)(a - b). a^4- (a - b)^4$
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Answer
$ a^4-(a-b)^4 $
$ =\left(a^2\right)^2-\left[\left(a^2-b^2\right)\right]^2 $
$ =\left[a^2+(a-b)^2\right]\left[a^2-(a-b)^2\right] $
$ =\left[a^2+a^2+b^2-2 a b\right]\left[a^2-\left(a^2-\left(a^2+b^2-2 a b\right)\right]\right. $
$ =\left[2 a^2+b^2-2 a b\right]\left[-b^2+2 a b\right] $
$ =\left(2 a^2+b^2-2 a b\right)(2 a b-b) $
$ =\left(a^2\right)^2-\left[\left(a^2-b^2\right)\right]^2 $
$ =\left[a^2+(a-b)^2\right]\left[a^2-(a-b)^2\right] $
$ =\left[a^2+a^2+b^2-2 a b\right]\left[a^2-\left(a^2-\left(a^2+b^2-2 a b\right)\right]\right. $
$ =\left[2 a^2+b^2-2 a b\right]\left[-b^2+2 a b\right] $
$ =\left(2 a^2+b^2-2 a b\right)(2 a b-b) $
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