In Exercises 37–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x) = 3x^4 - 2x^3 - 8x + 5

Descartes's Rule of Signs is a mathematical theorem that provides a way to determine the number of positive and negative real roots of a polynomial function. It states that the number of positive real zeros of a polynomial is equal to the number of sign changes between consecutive non-zero coefficients, or less than that by an even number. Similarly, for negative real zeros, one evaluates the polynomial at -x and counts the sign changes.

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable x is f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where n is a non-negative integer and a_n is not zero. Understanding the structure of polynomial functions is essential for applying Descartes's Rule of Signs effectively.

Sign changes refer to the transitions between positive and negative values in a sequence of numbers. In the context of polynomials, analyzing the coefficients of the polynomial allows us to count how many times the signs of these coefficients change. This count is crucial for applying Descartes's Rule of Signs to determine the potential number of positive and negative real zeros of the polynomial.