Descartes's Rule of Signs: Definition & Example

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Finding the roots of a polynomial is a common task in math and science applications. In this lesson, we use a method called Descartes's Rule of Signs to predict the number of roots we expect to find.

The Rule of Signs

The arena is packed! We hear Rene Day Cart (no relation to the famous mathematician-philosopher Rene Descartes) announcing the play-by-play: '… takes the puck, skates to the left, changes to the right, changes to the left, shoots and …'

In this lesson, we describe polynomials like Day Cart's play-by-play. Instead of reporting lefts and rights, we keep track of sign changes. The number of sign changes is related to the number of roots. We will start with a simple example and gradually build up our skills. By the end of the lesson, we will see how Descartes's rule of signs predicts the number of real roots to be at most equal to the number of sign changes.

Real Roots

Just for fun, let's do a sportscast!

A curve with three real roots

From the left, the curve climbs and crosses the horizontal axis (the x-axis). The curve continues to climb until it reaches a positive peak and then descends, crossing the vertical axis (the y-axis is the dashed vertical line). As the curve continues to move to the right, we see it crossing the x-axis, dipping below the x-axis and then climbing again where it once again crosses the x-axis.

The places on the x-axis where the curve crosses are the roots of the equation. On the right of the vertical axis, the roots are positive. See the blue circles?

For this curve, there is also a root to the left of the y-axis. This root has a negative value. See the green circle?

Finding the roots usually involves factoring or plotting a polynomial. What if we could learn something about the roots without factoring or plotting? Enter the Descartes's Rule of Signs. Counting the number of sign changes predicts the number of roots. We won't know where the curve crosses the x-axis but we will know how many roots to look for.

The equation for our example curve is

y = x^3 + x^2 - 10x + 8

On the right-hand side, we have a polynomial in x. The x terms have been ordered from highest exponent to lowest. Looking at the signs of each term, we see plus, plus, minus and then plus. We started with a plus, next term was still plus, then CHANGED to minus and then CHANGED to plus.

Two sign changes. How many positive roots did we see in the plot? Two. But what about the one negative real root?

To find the number of negative roots, replace the x in the equation with -x.

x^3 + x^2 - 10x + 8 becomes (-x)^3 + (-x)^2 - 10(-x) + 8 which equals

-x^3 + x^2 + 10x + 8.

How many sign changes now? We started with minus, CHANGED to plus, next term was still plus and the last term a plus. Total number of sign changes: 1. Total number of negative roots: 1. Aha!

The Descartes's rule of signs tells us:

  • The number of positive real roots is at most equal to the number of sign changes. The number of roots could be less by multiples of two.
  • After replacing x with -x, the number of negative real roots is at most equal to the number of sign changes. Again, the number of roots could be less by multiples of two.

What is this 'multiples of two' idea?

The Multiples of Two

Let's do another example:

x^7 + x^6 - 9x^5 + x^4 + 24x^3 - 26x^2 - 16x + 24

A huge polynomial! Don't worry, no factoring; just counting sign changes.

The signs of the terms: plus, plus, minus, plus, plus, minus, plus, plus. We started with plus, then plus, CHANGED to minus, CHANGED back to plus, then plus, CHANGED to minus, CHANGED to plus, stayed at plus and ended with plus. A total of 4 signs changes. Thus, the number of positive real roots is 4 or 2 or 0. Do you see how the 'less by multiples of two' is used? How about the negative real roots?

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