Question

Como fazer a fatoração

$<var>x^{8}-1</var>$

Answer 1

$<var>x^{8}-1\\ (x^{4}-1)(x^{4}+1)\\ (x^{2}-1)(x^{2}+1)(x^{4}+1)\\ (x-1)(x+1)(x^{2}+1)(x^{4}+1)\\</var>$

Answer 2

 Saiba que $a^2 - b^2 = (a + b)(a - b)$. Então:

 

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