Question 4(iv)
Factorise
$x^{4} - (x - z)^{4}$

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Solution

$x^{4} - (x - z)^{4} = (x^{2})^{2} - [((x - z)^{2})^{2}] = [x^{2} - (x - z)^{2}] [x^{2} + (x - z)^{2}] = [x - (x - z)] [x + (x - z)] [x^{2} + (x - z)^{2}] = z (2 x - z) [x^{2} + x^{2} - 2 x z + z^{2}] = z (2 x - z) (2 x^{2} - 2 x z + z^{2})$

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