I am interested in obtaining an approximate solution of $$x^n(x - (a+1)) + a = 0$$ where $x$ is allowed to be complex and we want asymptotics as $n \to \infty$. I have been unable to find methods to deal with this since the perturbation theory of algebraic equations normally assumes a fixed order. What general methods are there for this kind of problem?
Checking on a graph plotter, there are 3 real solutions at approximately $1$, $-1$ and $a+1$. However we would expect $n+1$ complex solutions.
EDIT: after doing more graph plotting, the solutions appear to be: one real solution at roughly $a+1$ and $n$ solutions approximately on the circle of radius $1$ in the complex plane. $1$ is itself always a solution.