Descartes’ Rule of Signs: A Guide to Determining Real Roots

Finding the real roots of polynomials is one of the most difficult problems to solve, especially if modern numerical methods are not used. While the Rational Root Theorem and synthetic division can help, Descartes’ Rule of Signs provides a simple yet effective method for estimating the number of positive and negative real roots of a polynomial. In this blog, we’ll look at Descartes’ Rule of Signs and how it might aid in polynomial analysis.

What is Descartes’ Rule of Signs?

Descartes’ Rule of Signs is a mathematical principle that determines the number of positive and negative real roots of a polynomial. It works by counting sign changes in the polynomial’s coefficient sequence. Observing these sign shifts allows us to forecast the number of real roots but not their exact quantities without having to solve the polynomial directly.

The Rule Explained

The rule consists of two main parts: determining the number of positive real roots and the number of negative real roots.

1. Positive Real Roots:
To determine the number of positive real roots, use the following steps:

• Write the polynomial in standard form, with terms sorted in descending order of powers of x.
• Consider the signs of the polynomial’s coefficients.
• Count the number of sign changes between successive coefficients.
• If the coefficient sequence changes from positive to negative or negative to positive, this counts as one sign change.
• Descartes’ Rule states that the number of positive real roots is either equal to the number of sign shifts or less by an even number. This means that if there are 4 sign changes, the polynomial can have 4, 2, or 0 positive real roots

2. Negative Real Roots:
To get the number of negative real roots, use these steps:

• Replace  x with  −x in the polynomial.
• Write a new polynomial P(−x) and check the coefficients’ signs.
• Count the number of sign changes in the coefficient sequence of P(−x).
• As with positive real roots, the number of negative real roots will either be equal to the number of sign changes or less by an even number.

Example of Descartes’ Rule of Signs

Let’s take a polynomial and apply Descartes’ Rule of Signs:

Step 1: Determine the Number of Positive Real Roots.
The coefficients are 1, -6, 12, -18, and 9.
1,−6,12,−18,9.

Signs are as follows: +, −, +, −, +
There are four sign changes: + → − , – → + , + → − , and  −→+.
According to Descartes’ Rule, the number of positive real roots is either 4, 2, or zero.

Step 2: Determine the number of negative real roots.
Substitute −𝑥 for x in the polynomial:

The coefficients of P(−x) are 1, 6, 12, 18, and 9.

The signs are all +, indicating no sign changes.
Thus, the polynomial has no negative real roots.

Interpretation of Results
Descartes’ Rule of Signs applies to the polynomial

• It has 4, 2 , or 0 positive real roots.
• It has zero negative real roots.

While Descartes’ Rule does not specify the number of roots, it does present a range of possibilities that can serve as a good starting point for further research or numerical procedures.