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

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1. Identify the polynomial function: \( f(x) = 3x^4 - 2x^3 + 0x^2 - 8x + 5 \).
2. To find the possible number of positive real zeros, count the number of sign changes in \( f(x) \). The sequence of coefficients is \( 3, -2, 0, -8, 5 \).
3. Observe the sign changes: from \( 3 \) to \( -2 \) (1 change), from \( -2 \) to \( 0 \) (no change), from \( 0 \) to \( -8 \) (no change), and from \( -8 \) to \( 5 \) (1 change). Thus, there are 2 sign changes.
4. According to Descartes's Rule of Signs, the number of positive real zeros is either equal to the number of sign changes or less than it by an even number. Therefore, there could be 2 or 0 positive real zeros.
5. To find the possible number of negative real zeros, consider \( f(-x) \): \( f(-x) = 3(-x)^4 - 2(-x)^3 + 0(-x)^2 - 8(-x) + 5 = 3x^4 + 2x^3 - 8x + 5 \). Count the sign changes in \( f(-x) \): from \( 3 \) to \( 2 \) (no change), from \( 2 \) to \( 0 \) (no change), from \( 0 \) to \( -8 \) (1 change), and from \( -8 \) to \( 5 \) (1 change). Thus, there are 2 sign changes, indicating there could be 2 or 0 negative real zeros.

Key Concepts

Descartes's Rule of Signs

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.

Polynomial Functions

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

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.