1A. Factorize the polynomial $x^{5}+x+1$.
Solution. For the given expression, we have

$$
\begin{aligned}
x^{5}+x+1 & =x^{5}+x^{4}+x^{3}-x^{4}-x^{3}-x^{2}+x^{2}+x+1 \\
& =x^{3}\left(x^{2}+x+1\right)-x^{2}\left(x^{2}+x+1\right)+\left(x^{2}+x+1\right) \\
& =\left(x^{2}+x+1\right)\left(x^{3}-x^{2}+1\right)
\end{aligned}
$$