Answer :

We can write x^{8} – y^{8} = (x^{4})^{2} – (y^{4})^{2}

We know: a^{2} – b^{2} = (a + b)(a – b)

x^{8} – y^{8} = (x^{4} + y^{4})(x^{4} – y^{4})

Rewriting (x^{4} – y^{4}) as (x^{2})^{2} – (y^{2})^{2}, we have

x^{8} – y^{8} = (x^{4} + y^{4})[(x^{2})^{2} – (y^{2})^{2}]

⇒ x^{8} – y^{8} = (x^{4} + y^{4})(x^{2} + y^{2})(x^{2} – y^{2})

Using the above identity to factorize (x^{2} – y^{2}), we have

x^{8} – y^{8} = (x^{4} + y^{4})(x^{2} + y^{2})(x – y)(x + y)

∴ x^{8} – y^{8} = (x – y)(x + y)(x^{2} + y^{2})(x^{4} + y^{4})

Hence, the factors of x^{8} – y^{8} are (x – y), (x + y), (x^{2} + y^{2}) and (x^{4} + y^{4}).

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