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I know about Des Cartes' rule of signs. But that can't help very much find out the exact number of positive, negative or real roots of a single-variable polynomial equation. So my question is if there is any rule to find out the exact number.

F Nishat
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  • Using Sturm's sequence https://en.wikipedia.org/wiki/Sturm%27s_theorem you get the exact number (if you don't mind multiple roots). – Michal Adamaszek Dec 03 '19 at 12:06

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Not sure if this is what you are looking for, but there are several numerical methods for finding a set of disjoint intervals of the real line such that each interval contains exactly one root of a given polynomial and their union contains all real roots of the polynomial. For an overview of these "real root isolation" methods see this Wikipedia article.

gandalf61
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