What is the remainder when the expression $x^{11}+x^{7}+x^{4}+x^{2}+x$ is divided by $(x-1)$?
Our expression can be transformed as follows:

$$
\begin{array}{r}
x^{11}-1+x^{7}-1+x^{4}-1+x^{2}-1+x-1+5= \\
(x-1)\left(x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)+ \\
+(x-1)\left(x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)+ \\
+(x-1)\left(x^{3}+x^{2}+x+1\right)+ \\
+(x-1)(x+1)+ \\
+(x-1) \cdot 1+5= \\
=(x-1)\left(x^{10}+x^{9}+x^{8}+x^{7}+2 x^{6}+2 x^{5}+2 x^{4}+3 x^{3}+4 x+5\right)+5
\end{array}
$$

from which it is evident that the sought remainder is 5.

Remark: Many solutions utilized the general division of polynomials or the Bezout theorem.