Descartes' Rule of Signs Made Simple

Descartes' Rule of Signs is a useful way to guess how many positive and negative solutions (or zeros) a polynomial function might have. This can help us understand where the graph of the function might change direction or turn.

How to Use the Rule:

1. Finding Positive Real Zeros:

   • Look at the polynomial $f(x)$ and see how many times the signs change when you plug in positive numbers.
   • For example, in $f(x) = x^4 - 3x^3 + 2x^2$, the signs are (+, -, +). This shows there are two sign changes. So, there could be 2 or even 0 positive real zeros.

2. Finding Negative Real Zeros:

   • Now, use $f(-x)$ and count the sign changes to check for negative zeros.
   • For our example, when we calculate $f(-x)$ for $f(x) = x^4 - 3x^3 + 2x^2$, we get $f(-x) = x^4 + 3x^3 + 2x^2$. Here, there are no sign changes, which means there are no negative real zeros.

Estimating Turning Points:

• What are Turning Points?: Turning points are places where the graph changes direction.
• The most turning points a polynomial can have is one less than its degree (the highest power of x).
• So, if you find $k$ positive zeros and $m$ negative zeros, the possible number of turning points is $k + m - 1$.

Simple Example:

For a polynomial with a degree of 4, if we discover 2 positive zeros and 0 negative zeros, we would expect up to $2 + 0 - 1 = 1$ turning point.

In short, Descartes' Rule of Signs helps us see how a polynomial might behave and makes it easier to sketch its graph.