# Finding roots of $4$th degree conjugate reciprocal polynomial

I am developing a computer program and the following polynomial, of which I need to obtain the roots, turned up $$Ax^4 + Bx^3 + Cx^2 + \overline{B}x + \overline{A}, \quad \text{where } A, B,x \in \mathbb{C}, \quad C \in \mathbb{R}$$

I googled around for a bit, and it turns out such a polynomial is called a conjugate reciprocal polynomial. There seems to be some research people have done on the location of the roots of such a polynomial, but I was unable to find anything on how to actually find them. Does their exist any clever algorithm to find the roots of such a polynomial, taking advantage of the symmetry involved? I know that if $A$ and $B$ are real, for example, you can make a change of variable $x' = x + 1/x$, and the problem is reduced to solving only quadratics. I am looking for any kind of trick or cleverness, maybe an iterative method, that would be suitable in this situation.

• Did you find anything? I want the same thing. Commented Sep 30, 2015 at 6:55
• Unfortunately not (yet). Commented Oct 1, 2015 at 14:27