Find the continued product:
$(x - 1) (x + 1) (x^{2} + 1) (x^{4} + 1)$

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Solution

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

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