Descartes’ Rule of Signs

We know how many roots a polynomial has by the highest degree of the polynomial. Descartes’ Rule of Signs allows us to determine possibly how many positive real roots, possible negative real roots, and possible complex roots there are in a polynomial.

Descartes’ rule works by counting the number of sign changes there are between terms. We them count the number of sign changes that occur when we substitute in (-x).

Example:

Given the polynomial:

First we look at the sign changes between the terms. The first term is positive, the second term is negative the third term is positive and the fourth term is negative.

+ - + -

We see three sign changes. So there are possibly 3 real positive roots. Here’s the catch. Because as we know that complex roots come in pairs there is the possibility that some of the positive roots could be positive complex roots. So when we state that there are 3 positive roots. We modify it slightly by saying there are 3 or 1 possible positive roots, but there is at least 1 positive real root. Essentially what we do is keep-subtracting 2 until we get a 1 or a 0. For example if we had 9 possible positive roots, we would say there are possibly 9, 7, 5, 3, or1 possible positive real roots.

Now let’s look at the negative side. We first take the polynomial and substitute in g(–x) for g(x).

Looking at which results in : . Now looking at the sign changes we get:

+ - - -

In this case we see only 1 sign change. So we know there is exactly one negative real root. (If we look at the graph it will cross the x – axis on the negative, or left side)

We have: possible positive real root: 3 or 1

Possible negative real root: 1

What about complex roots? We know there are exactly 4 roots. So now we look at the possibilities. If there are 3 positive roots and 1 negative roots, then there can be 0 complex roots. If there is 1 positive root and 1 negative root, then there must be 2 complex roots.

Example:

Given the polynomial:

First we look at the positive and see that there is only 1 sign change. So there must be exactly 1 positive real root. Looking at the negative side we see: , there are 3 sign changes. So there are 3 or 1 possible negative real roots.

We have: possible positive real roots: 1

Possible negative real roots: 3 or 1

Possible complex roots: 0, 2

When we do look at the graph we see that there are 3 negative roots and 1 positive root.

Hint: When substituting in (-x) into the original polynomial you should notice that when the exponent is even the sign will not change, but when the exponent is odd the sign does change. So you can simply change the polynomial by changing the signs on the odd exponent terms.