Factorise: $x^{4} - (y - z)^{4}$

(x^{2} + (y - z)^{2}) (x - y + z) (x + y - z)

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(x^{2} + (y + z)^{2}) (x - y - z) (x + y - z)

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(x^{2} + (y - z)^{2}) (x - y - z) (x - y - z)

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(x^{2} - (y - z)^{2}) (x + y - z) (x + y - z)

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Solution

The correct option is A $(x^{2} + (y - z)^{2}) (x - y + z) (x + y - z)$
Consider the given expression $x^{4} - (y - z)^{4}$ and solve it as shown below:

$x^{4} - (y - z)^{4} \Rightarrow {(x^{2})}^{2} - {((y - z)^{2})}^{2} \Rightarrow (x^{2} + (y - z)^{2}) (x^{2} - (y - z)^{2}) (∵ a^{2} - b^{2} = (a + b) (a - b)) \Rightarrow (x^{2} + (y - z)^{2}) [(x - (y - z)) (x + (y - z))] \Rightarrow (x^{2} + (y - z)^{2}) (x - y + z) (x + y - z)$

Hence, the factorisation of $x^{4} - (y - z)^{4}$ is $(x^{2} + (y - z)^{2}) (x - y + z) (x + y - z)$ .

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