Упр.33.57 ГДЗ Мордкович 7 класс (Алгебра)

6) x32 — (x — 1)(x + 1)(x2 + 1)(x4 + 1)(x8 + )(x16 + 1).

а)

$$ (a-b)(a+b)(a^2+b^2)(a^4+b^4)(a^8+b^8) $$
$$ =(a^2-b^2)(a^2+b^2)(a^4+b^4)(a^8+b^8) $$
$$ =(a^4-b^4)(a^4+b^4)(a^8+b^8) $$
$$ =(a^8-b^8)(a^8+b^8) $$
$$ =a^{16}-b^{16}. $$

б)

$$ x^{32}-(x-1)(x+1)(x^2+1)(x^4+1)(x^8+1)(x^{16}+1) $$
$$ =x^{32}-(x^2-1)(x^2+1)(x^4+1)(x^8+1)(x^{16}+1) $$
$$ =x^{32}-(x^4-1)(x^4+1)(x^8+1)(x^{16}+1) $$
$$ =x^{32}-(x^8-1)(x^8+1)(x^{16}+1) $$
$$ =x^{32}-(x^{16}-1)(x^{16}+1) $$
$$ =x^{32}-(x^{32}-1)=1. $$

Ответ

а) $$a^{16}-b^{16}$$; б) $$1$$.