When you study polynomials in Grade 12 Algebra II, one interesting idea you will learn about is Descartes' Rule of Signs.

This rule helps us figure out how many positive and negative real roots a polynomial has. It’s an important connection to the polynomial's degree. Let’s break it down to make it easier to understand.

Understanding Descartes' Rule of Signs

Descartes’ Rule of Signs tells us that we can find the number of positive roots of a polynomial by looking at the signs of the coefficients. Here’s how you can do it:

1. Put the Polynomial in Standard Form: Make sure your polynomial is written from the highest degree term to the lowest. For example:
   (P(x) = ax^n + bx^{n-1} + \ldots + k).

2. Count the Sign Changes: Look at the numbers (coefficients) in front of each term. Count how many times the sign changes as you move from the highest degree term to the last number (constant term). Every time the sign changes, it means there could be a positive root.

3. Find Possible Positive Roots: The number of possible positive roots can be the same as the number of sign changes, or it can be less by an even number. For example, if you see 3 sign changes, there could be 3, 1, or no positive roots.

Let’s see this with an example:

Example 1:

Think about the polynomial (P(x) = 2x^4 - 3x^3 + 5x^2 - 1).

• The coefficients are: 2, -3, 5, -1.
• The sign changes happen between:
  • 2 (positive) to -3 (negative) = 1 sign change.
  • -3 (negative) to 5 (positive) = 2 sign changes.
  • 5 (positive) to -1 (negative) = 3 sign changes.

So, there are 3 sign changes. This means the polynomial could have 3, 1, or no positive roots.

How to Find Negative Roots

To figure out how many negative roots a polynomial has, you can replace (x) with (-x) and then check the signs again. Here’s how:

1. Substitute: Replace (x) with (-x), which gives you (P(-x)).
2. Count the Sign Changes: Look at the signs of the coefficients for this new polynomial.

Example 2:

Using the same polynomial, (P(x) = 2x^4 - 3x^3 + 5x^2 - 1), we find:

$P(-x) = 2(-x)^4 - 3(-x)^3 + 5(-x)^2 - 1 = 2x^4 + 3x^3 + 5x^2 - 1$

The coefficients here are: 2, 3, 5, -1.

• The sign changes here are:
  • 2 (positive) to 3 (positive) = 0 sign change.
  • 3 (positive) to 5 (positive) = still 0.
  • 5 (positive) to -1 (negative) = 1 sign change.

Since there is 1 sign change, this means there is exactly 1 negative root.

The Link to Polynomial Degree

The degree of a polynomial is related to the number of its roots. A polynomial with degree (n) can have up to (n) real roots, which include both positive and negative roots.

1. For example, if your polynomial has a degree of 4, the total of positive and negative roots can be:

   • 3 positive and 1 negative.
   • 2 positive and 2 negative.
   • And so on.

2. Keep in mind that some roots might be complex or repeated. The rule doesn’t tell us about those kinds.

In summary, Descartes' Rule of Signs is a great tool to help us understand how polynomial functions behave with their roots while connecting back to the polynomial degree. The more you practice this, the clearer everything will become!