I have a third-grade polynomial of the form $Ax^3+Bx+C$ and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this:
$0=Ax^3+Bx+C$
and for a given root $X_0$:
$X_0=(-AX_0^3-C)/B$
I then guess that $X_0 = 3$ (just an example), then find a new value of $X$ until it converges to something near $3.54$. Sometimes when I'm not lucky it diverges.
I assume there must be a criterion to decide whether or not $X_0$ is a good approximation in the sense that it will converge to the root.