Understanding Descartes' Rule of Signs

Descartes' Rule of Signs is a handy tool for figuring out how many positive and negative roots a polynomial has.

This means we can find out how many roots are positive or negative without actually solving the polynomial.

Let’s look at how to do this in simple steps.

Finding Positive Roots

To find out how many positive roots a polynomial has, do these easy steps:

1. Put the polynomial in standard form: This just means writing the terms from highest degree to lowest degree.

2. Count the sign changes: Check the numbers (coefficients) in each term. Every time you see a change from a positive number to a negative one, or from negative to positive, count that as a sign change.

For example, let’s use this polynomial:

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

• The coefficients are: +1 (for $x^4$), -3 (for $-3x^3$), +2 (for $+2x^2$), and +5 (the constant term).
• Now, let’s count the sign changes:
  • From +1 to -3 (that’s 1 change)
  • From -3 to +2 (no change)

So, we have 1 sign change. This tells us there is 1 positive root.

Finding Negative Roots

Next, to find the negative roots, we need to look at the polynomial by using $-x$.

1. Substitute $-x$ into the polynomial.
2. Count the sign changes in this new polynomial.

For our example, we change it to:

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

Now, let’s look at the new coefficients: +1 (for $x^4$), +3 (for $3x^3$), +2 (for $2x^2$), and +5 (the constant).

Here, there are no sign changes. This means we have 0 negative roots.

In Summary

So, Descartes' Rule of Signs helps us quickly estimate how many positive and negative roots a polynomial has.

This makes solving these kinds of problems easier and faster!