Factorise: $x^{8} - 1 = ?$

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(x^{2} + 1 + \sqrt{2} x) (x^{2} + 1 - \sqrt{2} x) (x + 1) (x - 1)

(x^{2} + 1 + \sqrt{2} x) (x^{2} + 1 - \sqrt{2} x) (x + 1) (x - 2)

(x^{2} + 1 + \sqrt{2} x) (x^{2} + 1 - \sqrt{2} x) (x + 2) (x - 1)

(x^{2} + 1 + \sqrt{2} x) (x^{2} + 1 - \sqrt{2} x) (x + 1) (x - 3)

answer is A.

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Detailed Solution

We may also format it as

x^{8} - 1 = {(x^{4})}^{2} - 1^{2}

According to the equation

a^{2} - b^{2} = (a + b) (a - b)

Here we get

x^{8} - 1 = (x^{4} + 1) (x^{4} - 1)

x^{8} - 1 = (x^{4} + 1) ({(x^{2})}^{2} - 1^{2})

x^{8} - 1 = (x^{4} + 1) (x^{2} + 1) (x^{2} - 1)

Hence, we get

x^{8} - 1 = (x^{4} + 1) (x^{2} + 1) (x^{2} - 1)

Adding more terms and applying the formula

x^{8} - 1 = [{(x^{2})}^{2} + 1^{2} + {2 x}^{2} - {2 x}^{2}] (x^{2} - 1) (x + 1) (x - 1)

On further simplification

x^{8} - 1 = [({(x^{2})}^{2} + 1^{2} + 2 x^{2}) - {2 x}^{2}] (x^{2} + 1) (x + 1) (x - 1)

The result of the formula is

x^{8} - 1 = (x^{2} + 1 + \sqrt{2} x) (x^{2} + 1 - \sqrt{2} x) (x + 1) (x - 1)

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