Prove that system of equation implies statement How to prove that
$$
\begin{cases}
x_1 + x_2 + x_3 & = 0 \\
x_1x_2 + x_2x_3 + x_3x_1 & = p \\
x_1x_2x_3 & = -q \\
x_1 & = 1/x_2 + 1/x_3
\end{cases}
$$
implies
$$
q^3 + pq + q = 0\,\,?
$$

Denote the equations by $(1),\ldots ,(4)$. Then $(4)$ says $x_1x_2x_3=x_2+x_3$ and $(3)$ says $x_1x_2x_3=-q$. This gives $x_3=-x_2-q$. Substitute this into $(1)$. This gives 
$x_1=q$.
 Then $q\cdot (2)-(3)$ gives $-q(p+q^2+1)=0$, or 
$$q^3+pq+q=0.$$
