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)=5x³−3x²+3x−1

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1. Identify the polynomial function: $f(x)=5x^3-3x^2+3x-1$.
2. Apply Descartes's Rule of Signs to find the possible number of positive real zeros by counting the sign changes in $f(x)$: Look at the coefficients sequence $5,\; -3,\; 3,\; -1$ and count how many times the sign changes from positive to negative or vice versa.
3. Apply Descartes's Rule of Signs to find the possible number of negative real zeros by evaluating $f(-x)$: Substitute $-x$ into the function to get $f(-x)\; =\; 5(-x)^3\; -\; 3(-x)^2\; +\; 3(-x)\; -\; 1$, simplify the expression, and then count the sign changes in the coefficients of $f(-x)$.
4. Interpret the results: The number of positive real zeros is equal to the number of sign changes in $f(x)$ or less than that by an even number (e.g., if there are 3 sign changes, possible positive zeros are 3 or 1). Similarly, the number of negative real zeros is equal to the number of sign changes in $f(-x)$ or less than that by an even number.
5. Summarize the possible numbers of positive and negative real zeros based on the counts from the previous steps, keeping in mind that the total number of real zeros cannot exceed the degree of the polynomial.

Key Concepts

Descartes's Rule of Signs

Descartes's Rule of Signs is a method to determine the possible number of positive and negative real zeros of a polynomial. It involves counting the number of sign changes in the coefficients of f(x) for positive zeros and f(-x) for negative zeros. The actual number of positive or negative zeros is either equal to the number of sign changes or less than it by an even number.

Polynomial Functions and Their Zeros

A polynomial function is an expression involving variables raised to whole-number exponents with coefficients. The zeros of a polynomial are the values of x that make the function equal to zero. Understanding the degree and coefficients helps in analyzing the behavior and number of possible roots.

Evaluating f(-x) to Find Negative Zeros

To apply Descartes's Rule of Signs for negative zeros, substitute -x into the polynomial to get f(-x). This changes the signs of terms with odd powers of x. Counting sign changes in f(-x) reveals the possible number of negative real zeros, similar to the process for positive zeros.