In Exercises 33–38, use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=5x^3−3x^2+3x−1

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1. Identify the polynomial function: \( f(x) = 5x^3 - 3x^2 + 3x - 1 \).
2. To find the possible number of positive real zeros, examine the sign changes in \( f(x) \).
3. List the coefficients of \( f(x) \): 5, -3, 3, -1.
4. Count the sign changes between consecutive coefficients: 5 to -3 (change), -3 to 3 (change), 3 to -1 (change).
5. According to Descartes's Rule of Signs, the number of positive real zeros is equal to the number of sign changes or less than that by an even number: 3, 1, or 0 positive 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 is equal to the number of sign changes between consecutive non-zero coefficients of the polynomial, or less than that by an even number. For negative real zeros, the rule is applied to the polynomial evaluated at -x, analyzing the sign changes in that expression.

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 a_n is not zero. The degree of the polynomial, determined by the highest power of x, indicates the maximum number of real roots it can have, which is crucial for applying Descartes's Rule of Signs.

Sign Changes

Sign changes refer to the transitions between positive and negative values in a sequence of numbers. In the context of Descartes's Rule of Signs, identifying sign changes in the coefficients of a polynomial helps determine the potential number of positive and negative real zeros. For example, if a polynomial has coefficients of alternating signs, it indicates multiple roots, while a consistent sign suggests fewer or no roots.