\( 1 \leftarrow \quad \) Use Descartes's rule of signs to discuss the possibilities for the roots of the equation. Do not solve the equation.

Descartes's rule of signs is a neat little tool that helps us predict the number of positive and negative real roots of a polynomial. For positive roots, we count the number of sign changes in the polynomial when the variable appears as is. For negative roots, we substitute \( x \) with \( -x \) and then count the sign changes again. This gives us a glimpse into how many roots there are and can alert us to the possible number of solutions without having to solve the equation outright! To illustrate, suppose we have a polynomial \( p(x) = x^3 - 3x^2 + 4 \). Looking for positive roots, we find that \( p(1) = 2 \) (positive) and \( p(2) = 0 \) (switching signs). This indicates there’s one or a maximum of one positive root. For negative roots, substituting \( x \) with \( -x \) gives us a new behavior to analyze, revealing more about root possibilities. It's like piecing together a puzzle without needing to see the full picture!