2. Factorize the polynomial $x^{3}-7 x+6$.

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Solution. We have

$$
\begin{aligned}
x^{3}-7 x+6= & x^{3}-x-6 x+6 \\
& =x\left(x^{2}-1\right)-6(x-1) \\
& =x(x-1)(x+1)-6(x-1) \\
& =(x-1)[x(x+1)-6] \\
& =(x-1)\left(x^{2}+x-6\right) \\
& =(x-1)\left(x^{2}-4+x-2\right) \\
& =(x-1)[(x-2)(x+2)+x-2] \\
& =(x-1) (x-2)(x+2+1) \\
& =(x-1)(x-2)(x+3)
\end{aligned}
$$