Describe the Descartes' rule of signs for polynomials.

Descartes' rule of signs is a method to determine the number of positive and negative roots of a polynomial.

To apply Descartes' rule of signs, we need to count the number of sign changes in the coefficients of the polynomial when it is written in standard form. For example, the polynomial f(x) = 3x^3 - 2x^2 + 5x - 1 has three sign changes: from positive to negative in the coefficient of x^2, from negative to positive in the coefficient of x, and from positive to negative in the constant term.

The number of positive roots of the polynomial is either equal to the number of sign changes or less than it by an even number. In the example above, the number of positive roots is either 3 or 1. Similarly, the number of negative roots is either equal to the number of sign changes or less than it by an even number. In the example above, the number of negative roots is either 1 or 3.

If the number of positive roots and negative roots are not equal to the degree of the polynomial, then there must be some roots that are complex or repeated. For example, if a cubic polynomial has two positive roots and one negative root, then the third root must be complex or repeated.

Descartes' rule of signs is a useful tool for determining the possible number of roots of a polynomial without actually finding them. However, it does not tell us the exact values of the roots or their multiplicities.