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.