Can a polynomial equation always be manipulated to give a recurrence formula? Let $p(x)$ be a real (or maybe complex) polynomial. Suppose we wish to (numerically) solve $p(x) = 0$. This can be done for example with Newton's method of course, but I was thinking about if you "solve $x$" from the equation somehow and then start iterating. For example, if $p(x) = x^5 + x^2 -1$, you can solve 
$$x=\frac{1}{\sqrt{x^3+1}}$$
and starting from $x_0=1$ recursing (plugging $x$ back to the formula $\frac{1}{\sqrt{x^3+1}}$) gives $x = 0.80873...$
I wonder if it's always possible to derive a recurrence equation from a polynomial to find out its roots starting from some initial value? Well, if you start from the root it stays in the root, but maybe in a way that there is some open set around the root such that the root pulls it into itself.

it depends on the right hand side of x=g(x). If g(x) is a contraction mapping or if the initial iteration is within a compact set where g sends to itself or at least if futher iterations lie within such a compact set. 
