X8 1=x42 12=x4+1x4 1 #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; # ...

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Polynomials

Factorise:x8-...

Question

Factorise:
x⁸ – 1

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Solution

$x^{8} - 1 = {(x^{4})}^{2} - 1^{2} = (x^{4} + 1) (x^{4} - 1) [a^{2} - b^{2} = (a + b) (a - b)] = (x^{4} + 1) [{(x^{2})}^{2} - 1^{2}]$

$= (x^{4} + 1) (x^{2} + 1) (x^{2} - 1) [a^{2} - b^{2} = (a + b) (a - b)] = (x^{4} + 1) (x^{2} + 1) {(x^{2} - 1)}^{2} = (x^{4} + 1) (x^{2} + 1) (x + 1) (x - 1) [a^{2} - b^{2} = (a + b) (a - b)]$

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