I am stuck at a algorithm problem. I was assigned the following problem: Given a finite Taylor expansion in some degree $n$: $$\text{TE} = \sum_{j=0}^{n}{c_j}{x^j}$$ with all real coefficients $c_j$ known except for a single coefficient $c_k$, I need to find the minimum value for $c_k$ such that $\text{TE} = 0$ for some value of $x$. So I have tried to isolate it, which gives a rational function in $x$: $$c_k = \frac{\sum_{j \neq k}c_jx^j}{x^k}$$ However now I am stuck. Normally to find the global minimum of a rational function you differentiate the function then set the derivative to $0$ and find all critical points. But when the degree is high, I don't know about a stable method to find all the roots of this rational function derivative. We were only taught the Newton's method so far, but it is unstable. So is there a stable numerical method that can find all the roots of the function in a low time complexity? We are not allowed to exceed quadratic time complexity.
Possibly I was told that there exists an algorithm called VCA (Vincent Collins Akritas) for polynomials, but it requires the Budan's theorem to determine the number of roots within an interval. Is there an extension for rational functions? This would allow an $O(n \log n)$ algorithm.