Question: The following polynomial is given: \[ g(x) = x^3 - 7x^2 - 6x + 72 \] Use Descartes’ Rule of…

The following polynomial is given:

\[ g(x) = x^3 - 7x^2 - 6x + 72 \]

Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros. Be sure to include all possible values.

Number of Positive Real Zeros: _____

Number of Negative Real Zeros: _____

Solution

First, consider the polynomial: \[ g(x) = x^3 - 7x^2 - 6x + 72 \] Count the number of sign changes in \( g(x) \) to determine the possible number of positive real zeros. There are two sign changes. Therefore, the number of positive real zeros is: \[ \begin{cases} 2 & \\ 0 & \end{cases} \] Next, consider \( g(-x) \): \[ g(-x) = -x^3 - 7x^2 + 6x + 72 \] Count the number of sign changes in \( g(-x) \) to determine the possible number of negative real zeros. There is one sign change. Therefore, the number of negative real zeros is: 1