Polya's Theorem

Polya’s Theorem provides a systematic way to determine whether a polynomial equation has roots within a certain interval, typically by analyzing the signs and behavior of the polynomial and its derivatives. It extends the idea behind the Intermediate Value Theorem, helping to identify intervals where solutions may exist without solving the equation directly. Essentially, it offers criteria—based on the polynomial’s coefficients and the shape of its graph—to assess the presence and number of roots within a specified range, making it a useful tool in understanding the qualitative behavior of polynomial functions.