Budan's theorem

Budan's theorem is a result in algebra that provides a method for determining the number of real roots of a polynomial within a specific interval. Specifically, it relates to the evaluation of the signs of the polynomial and its derivatives at the endpoints of the interval. The theorem can be stated as follows: 1. Consider a polynomial \( P(x) \) of degree \( n \) and its derivative \( P'(x) \).