What mathematical principle determines the maximum number of positive real roots for a polynomial?

The maximum number of positive real roots for a polynomial is determined by Descartes' Rule of Signs. This principle states that the number of positive real roots of a polynomial can be found by counting the number of sign changes in the coefficients of the polynomial when written in standard form. Each sign change indicates a potential positive root, and the actual number of positive roots can be either equal to this count or less than it by an even integer.

For example, if a polynomial has three sign changes, it can have up to three positive real roots, one positive real root, or none at all. This rule is particularly useful as it gives insights into the behavior of polynomials without needing to find the roots explicitly.

The other choices, while important in their own right, serve different purposes; the Fundamental Theorem of Algebra states that a polynomial equation will have a number of roots equal to its degree, but it does not specifically address how many of those roots are positive. The Quadratic Formula provides a method to find the roots of quadratic equations, and the Binomial Theorem expands expressions raised to a power, but neither of these directly relates to the maximum number of positive real roots of a polynomial.