Разложите на множители многочлен:
а) $x^8 + x^4 - 2$;
б) $a^5 - a^2 - a - 1$;
в) $n^4 + 4$;
г) $n^4 + n^2 + 1$.
$x^8 + x^4 - 2 = x^8 + x^4 - 1 - 1 = (x^8 - 1) + (x^4 - 1) = (x^4 - 1)(x^4 + 1) + (x^2 - 1)(x^2 + 1) = (x^2 - 1)(x^2 + 1)(x^4 + 1) + (x - 1)(x + 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1)(x^2 + 1) + (x - 1)(x + 1)(x^2 + 1) = (x - 1)(x + 1)(x^2 + 1)(x^4 + 1 + 1) = (x - 1)(x + 1)(x^2 + 1)(x^4 + 2)$
$a^5 - a^2 - a - 1 = (a^5 - a) - (a^2 + 1) = a(a^4 - 1) - (a^2 + 1) = a(a^2 - 1)(a^2 + 1) - (a^2 + 1) = (a^2 + 1)(a(a^2 - 1) - 1) = (a^2 + 1)(a^3 - a - 1)$
$n^4 + 4 = n^4 + 4 - 4n^2 + 4n^2 = (n^4 + 2n^2 + 4) - 4n^2 = (n^2 + 2)^2 - (2n)^2 = (n^2 + 2 - 2n)(n^2 + 2 + 2n)$
$n^4 + n^2 + 1 = n^4 + n^2 + 1 + n^2 - n^2 = (n^4 + 2n^2 + 1) - n^2 = (n^2 + 1)^2 - n^2 = (n^2 + 1 - n)(n^2 + 1 + n)$
Пожауйста, оцените решение
