Expand: $(x - 1) (x + 1) (x^{2} + 1) (x^{4} + 1)$

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Solution

Consider $(x - 1) (x + 1) (x^{2} + 1) (x^{4} + 1)$
Now, $(x - 1) (x + 1) = x^{2} - 1^{2}$ ............. $∵ (a + b) (a - b) = (a^{2} - b^{2})$
$⟹ (x - 1) (x + 1) (x^{2} + 1) (x^{4} + 1) = (x^{2} - 1) (x^{2} + 1) (x^{4} + 1)$
Taking $a = x^{2}, b = 1$ in $(a + b) (a - b) = (a^{2} - b^{2})$
$⟹ (x - 1) (x + 1) (x^{2} + 1) (x^{4} + 1) = ((x^{2})^{2} - (1)^{2}) (x^{4} + 1)$
$= (x^{4} - 1) (x^{4} + 1)$
We have, $(x^{4} - 1) (x^{4} + 1)$ = $(x^{4})^{2} - (1)^{2}$ ....... [Again by the formula]
$⟹ (x - 1) (x + 1) (x^{2} + 1) (x^{4} + 1) = (x^{8} - 1)$
Hence, the answer is $x^{8} - 1$

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