Problem 50 - College Algebra

Question

Describe how to use Descartes's Rule of Signs to determine the possible number of positive real zeros of a polynomial function.

Step-by-Step Solution

Answer

Descartes's Rule of Signs can help us to get the possible number of positive real zeros of a polynomial function. For a polynomial, count the number of sign changes between consecutive non-zero coefficients, which gives a number. The possible number of positive real zeros is this number or less by an even number.

Step 1: Understand Descartes's Rule of Signs

Descartes's Rule of Signs provides a method to determine the possible number of positive real zeros of a polynomial function. The rule states that the number of positive real zeros in a polynomial function \(f(x)\) is the same as the number of sign changes between consecutive non-zero coefficients of \(f(x)\), or is less than this by an even number. It only provides us with the possible numbers, not the exact number.

Step 2: Identify the Polynomial

Decide on a polynomial for which you wish to find the possible number of positive real zeros. Let us consider an example of a polynomial equation: \(f(x) = 2x^4 - 3x^3 + x^2 - 2\).

Step 3: Count the Number of Sign Changes

Observe the polynomial and count the number of sign changes between consecutive non-zero coefficients. In our example \(f(x) = 2x^4 - 3x^3 + x^2 - 2\), we have 2 sign changes, from \(2x^4\) to \(-3x^3\) and from \(x^2\) to \(-2\).

Step 4: Apply Descartes's Rule of Signs

Applied rule says that the number of positive real zeros of the function is either equal to the number of sign changes or less than this by an even number. In our case, we've 2 sign changes, so we can have 2 or 0 positive real zeros.