Problem 67 - 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

To use Descartes' Rule of Signs, first write the polynomial function in standard form and count the number of times the signs change from coefficient to coefficient. This gives you the maximum number of positive real zeros which could be less by a multiple of two.

Step 1: Understand Descartes' Rule of Signs

Descartes' Rule of Signs states that the number of positive real zeros in a polynomial function can be found by counting the number of sign changes in the function when it is written in standard form. Note, however, that Descartes' Rule of Signs provides the maximum number of positive real zeros; the actual number may be less by a multiple of two.

Step 2: Write the Polynomial in Standard Form

Write the polynomial function in standard form. The standard form of a polynomial function is a mathematical expression arranged by descending powers of the variable. For example, the polynomial \(4x^3 - 2x^2 + 7x -3\) is already in standard form.

Step 3: Count the Number of Sign Changes

Find variations from positive to negative coefficients (or vice versa). Each change in sign represents a possible real positive zero. For example, in the polynomial \(4x^3 - 2x^2 + 7x -3\), there is one sign change, from negative \(-2x^2\) to positive \(7x\). Therefore, there's a maximum of one positive real zero in this polynomial function.

Step 4: Consider Additional Possibilities

Depending on the number of sign changes, there might be fewer real positive zeroes, since the actual number could be less than the number of sign changes by a multiple of two. In this case, there's only one sign change, so the polynomial could have one or zero positive real zeros. If there were two sign changes, the polynomial could have two or zero positive real zeros, and so forth.