You use Budan's theorem https://en.wikipedia.org/wiki/Budan%27s_theorem https://en.wikipedia.org/wiki/Budan%27s_theorem to certify that there's exactly one (or

You use Budan's theorem https://en.wikipedia.org/wiki/Budan%27s_theorem https://en.wikipedia.org/wiki/Budan%27s_theorem to certify that there's exactly one (or exactly zero) real root in the interval (r - ɛ, r + ɛ]. As long as your guess r is within ɛ of a root, this allows distinguishing between real and complex roots even if you don't have the exact value. (There are instances where Budan's theorem cannot give an answer, e.g. most obviously if there's more than one root in the interval.)