To motivate some ideas in my research, I've been looking at the expected number of real roots of random polynomials (and their derivatives). In doing so I have noticed an issue/bug with** fsolve** and **RootFinding[Isolate]**. One of the polynomials I came upon was

f(x) = -32829/50000-(9277/50000)*x-(37251/20000)*x^2-(6101/6250)*x^3-(47777/20000)*x^4+(291213/50000)*x^5.

We know that f(x) has at least 1 real root and, in fact, graphing shows that f(x) has exactly 1 real root (~1.018). However, **fsolve(f)** and **Isolate(f)** both return no real roots. On the other hand, **Isolate(f,method=RC)** correctly returns the root near 1.018. I know that **fsolve**'s details page says "It may not return all roots for exceptionally ill-conditioned polynomials", though this system does not seem especially ill-conditioned. Moreover, **Isolate**'s help page says confidently "All significant digits returned by the program are correct, and *unlike purely numerical methods no roots are ever lost*, although repeated roots are discarded" which is clearly not the case here. It also seems interesting that the **RealSolving** package used by **Isolate(f,method=RS)** (default method) misses the root while the **RegularChains** package used by **Isolate(f,method=RC)** correctly finds the root.

All-in-all, I am not sure what to make of this. Is this an issue which has been fixed in more recent incarnations of **fsolve** or **Isolate**? Is this a persistent problem? Is there a theoretical reason why the root is being missed, particularly for **Isolate**?

Any help or insight would be greatly appreciated.