There's an incredibly simple result of Sturm that allows one to compute intervals bounding the real roots of polynomials. https://en.wikipedia.org/wiki/Sturm%2

There's an incredibly simple result of Sturm that allows one to compute intervals bounding the real roots of polynomials. https://en.wikipedia.org/wiki/Sturm%27s_theorem https://en.wikipedia.org/wiki/Sturm%27s_theorem That said, if you're working with finite precision, then there's an interesting question somewhere here -- given a polynomial with a certain number of real roots, how precisely do you need to know the coefficients before the answer changes?