8. The remainder when $x^{20001}-1$ is divided by $x^{4}+x^{3}+2 x^{2}+x+1$ is

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8. $-2 x^{3}-x^{2}-2 x-1$

Let the remainder be $a x^{3}+b x^{2}+c x+d, x^{4}+x^{3}+2 x^{2}+x+1=\left(x^{2}+1\right)\left(x^{2}+x+1\right)$
So $\left\{\begin{array}{l}-a \mathrm{i}-b+c \mathrm{i}+d=\mathrm{i}^{2000}-1 \\ a \omega^{3}+b \omega^{2}+c \omega+d=\omega^{2000}-1\end{array}\right.$
Solving, the remainder is $-2 x^{3}-x^{2}-2 x-1$