32.21. Let $x_{1}, x_{2}, x_{3}$ be the roots of the polynomial $x^{3}+p x+q$. Compute $s_{n}=x_{1}^{n}+x_{2}^{n}+x_{3}^{n}$ for $n=1,2, \ldots, 10$.
32.21. The equality $x_{i}^{n+3}+p x_{i}^{n+1}+q x_{i}^{n}=0$ shows that the recurrence relation $s_{n+3}+p s_{n+1}+q s_{n}=0$ holds. It is also clear that $s_{0}=3$ and $s_{1}=0$. Moreover, $s_{-1}=\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}=\frac{x_{2} x_{3}+x_{1} x_{3}+x_{1} x_{2}}{x_{1} x_{2} x_{3}}=$ $=-\frac{p}{q}$. Now we can compute $s_{n}$ using the known values of $s_{-1}, s_{0}, s_{1}$ and the recurrence relation. As a result, we get $s_{2}=-2 p, s_{3}=-3 q, s_{4}=2 p^{2}$, $s_{5}=5 p q, s_{6}=-2 p^{3}+3 q^{2}$, $s_{7}=-7 p^{2} q, s_{8}=2 p^{4}-8 p q^{2}, s_{9}=9 p^{3} q-3 q^{3}, s_{10}=-2 p^{5}+15 p^{2} q^{2}$.