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
$$
q^3 + pq + q = 0\,\,?
$$

According to the first three equations, $x_1, x_2, x_3$ are solutions of 
$$x^3 +px + q = 0$$
The fourth equation can be translated into $x_1x_2x_3 = x_3 + x_2$, i.e. $-q = -x_1$, then $x_1 = q$,
Conclude by noting that $q$ solves $x^3 +px + q = 0$
