Find roots of polynomial with degree $ge 5$

During our research we came up with the problem of computing the root of a polynomial of degree $\ge 5$ exactly. The coefficients are, except for the linear and constant term, all non-negative and there are only terms with even degree. The only thing we know is that there formulas for a degree up to 4 and no formula for a higher degree, but is it possible to compute the roots of a higher-degree polynomial exactly, too? If so, what is the complexity?

Here is some more information:

The equation looks like $A(n) - x - a_0 = 0$ for some arbitrary integral $n$. Thereby, $A(0)=x$ and $A(i) = a_i \cdot A(i-1) \cdot (A(i-1)+b_i)$ for $i \in \{1,\ldots,n\}$. All the values $a_i,b_i$ are non-negative real numbers for $i \in \{1,\ldots,n\}$ whereas $a_0$ is arbitrary.