Descartes' Rule of Signs
2026-02-28 11:39 Diff
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Last updated on October 25, 2025

Descartes' rule of signs is an algebraic theorem that estimates the maximum number of positive and negative real roots of a polynomial equation. Over time, the rule has been extended to algorithms like Budan’s theorem and real-root isolation.

What is Descartes' Rule of Signs?

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Descartes' rule of signs helps determine the number of positive and negative real roots by counting sign changes in the terms of the polynomial. It states that,

\(f(x) = a_nx^n + a_{n-1}x^{n-1} + . . . + a_1x + a_0\)


Here, the single variable polynomial is written in standard form. 

Note: This rule only applies to polynomials having real coefficients. We need to avoid terms with 0 as a coefficient, when counting sign change.


The rules applied are:

  1. The number of positive real roots is equal to the sign changes in f(x) or less than that by an even number.

    For example, if there are 5 sign changes in f(x), then the number of positive roots will be:
    • 5 in case all roots are real and distinct, 
    • 3 (5 - 2), or
    • 1 (5 - 4)

  2. To find the number of negative real roots, substitute -x into the polynomial and count the sign changes.  

How to Apply Descartes' Rule of Signs?

Descartes' rule of signs can be applied using the following steps:

  1. Arrange the polynomials in standard form, with exponents, in descending order.
     
  2. Count sign changes in f(x) to estimate the maximum number of real roots.
     
  3. Substitute f(x) with f(-x) and count sign changes to estimate the maximum number of negative real roots.
     
  4. To find the number of complex/Imaginary roots, subtract total real roots (positive and negative) from the degree of the polynomial.

    Complex roots = degree of polynomial - (no. of positive roots + no. of negative roots)

Let’s apply these steps to an example.


Question: Apply Descartes' rule of signs to estimate the number of positive, negative, and imaginary roots of the given polynomial: \(f(x) = 2x^4 - 3x^3 - 5x^2 + 9x - 4 \)

  • Step 1: The polynomial is already in standard form, so we will write it as is.
    \(f(x) = 2x^4 - 3x^3 - 5x^2 + 9x - 4 \)
  • Step 2: Counting sign changes in f(x) 
    The signs of the coefficients are +, -, -, +, -

    Sign changes:

  1. + → –   (1 change)
  2. –  → –  (0 change)
  3. –  → +  (1 change)
  4. + → –   (1 change)

    There are 3 sign changes in f(x), so the total number of positive real roots is either 3 or 1, depending on the number of complex roots.

  • Step 3: Substitute -x in f(x) to find negative real roots
    \(f(-x) = 2(-x)^4 - 3(-x)^3 - 5(-x)^2 + 9(-x) - 4\)

    Upon simplification, we get
    \(2x^4 + 3x^3 - 5x^2 - 9x - 4\)

    Now counting sign changes:

  1. + → +   (0 changes)
  2. + → –    (1 change)
  3. – → –    (0 changes)
  4. – → –    (0 changes)

    The number of negative real roots is equal to the number of sign changes, which is 1.

  • Step 4: For finding the number of imaginary roots:
    The degree of the polynomial is 4, so the equation will have a total of 4 roots, including real, complex, and their multiplicity.

    Complex roots = degree of polynomial - (no. of positive roots + no. of negative roots)

    So, there can be two cases: 

  1. Number of positive real roots = 3
    Number of negative real roots = 1
    Number of complex roots = 4 - (3 + 1) = 0
    Or,
     
  2. Number of positive real roots = 1
    Number of negative real roots = 1
    Number of complex roots = 4 - (1 + 1) = 4 - 2 = 2

    So, 
    Number of positive real roots = 3 or 1
    Number of negative real roots = 1
    Number of complex roots = 0 or 2

Descartes' Rule of Signs Chart

While Descartes' rule of signs does not give us the exact number of roots, we can create a chart with the possible number of real roots. A few things to keep in mind while constructing this chart are:

  1. For polynomials with real coefficients, imaginary roots always occur in conjugate pairs, appearing in even numbers. This means if one root is a + bi, then the other must be a - bi.
  2. If there are 0 or 1 sign changes in f(x) or f(-x), then the number of negative or positive real roots is exact.
    Reason: It is so because subtracting an even number from 0 or 1 will give a negative value, and the number of roots cannot be a negative value.
  3. The total number of roots, including multiplicities, is equal to the polynomial’s degree. Therefore, the number of complex roots can be found by subtracting the sum of positive and negative real roots from the degree of the polynomial.

Following these facts, let’s construct Descartes' rule of signs using the same example discussed above: polynomial \(f(x) = 2x^4 - 3x^3 - 5x^2 + 9x - 4 \):

Positive Real Roots

Negative Real Roots

Complex (Imaginary) Roots

Total Roots

3 1 0 4 1 1 2 4

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Generalizations of Descartes' Rule of Signs

There are two generalizations of Descartes' rule of signs:

  1. Non-real roots

    According to the fundamental theorem of algebra, any polynomial of degree n has exactly n roots, whether real or complex.

    Non-real roots = n - (positive real roots + negative real roots)

  2. Fewnomial Theory

    Khovanski’s Fewnomial theory extends Descartes' rule of signs to include functions involving transcendental terms like exponentials and logarithms, etc. It suggests that the number of real roots depends more on the number of terms than on the degree of the expression.

    So, according to the Fewnomial theory, even if an equation has a higher degree, it can have fewer real roots depending on the number of terms present in the equation.

Budan’s Theorem & Real-Root Isolation

Descartes' rule is extended using the method of linear fractional transformation. This idea forms the basis of Budan’s theorem and Budan-Fourier theorem. These theorems are studied on higher level, and has many applications.
 

  • Such theorems allow us to estimate the number of real roots present within any specified interval.
     
  • These extensions are widely used in fast computer algorithms for root calculations.
     
  • Real-root isolation locates each root of a polynomial within a separate interval on the number line.
     
  • It ensures that each interval contains exactly one real root and does not overlap with intervals of other roots.

Common Mistakes and How to Avoid Them in Descartes' rule of signs

Given below is a summary of some common errors while applying Descartes’ rule of signs, along with their solutions for reference.
 

Real-Life Applications of Descartes' rule of signs

Descartes' rule of signs is used to estimate the number of positive and negative real roots of a polynomial equation. Here are some real-life applications of the rule:

  1. Mathematical education and proof validation.

    Descartes' rule is used in curriculum design and automated proof checking. It is also used to simplify steps in solving polynomials.

  2. Stability analysis in control engineering.

    Engineers use Descartes' rule to estimate the number of positive real roots to indicates whether the system is stable or unstable.

  3. Determining feasible economic equilibrium points.

    Economists model supply and demand, profit functions, and utility functions using polynomials. Descartes' rule helps determine the number of economically viable solutions that exist.

  4. Identifying real reaction steady states in chemical kinetics.

    In chemical kinetics, the rule indicates the number of physically possible concentrations or rates. This helps chemists predict how many stable reaction states are chemically feasible.

  5. To improve root-finding algorithms in computer algebra systems.

    Mathematical software programs like MATLAB, Mathematica, or Wolfram Alpha use symbolic algorithms for finding roots of polynomials. Descartes’ rule of signs helps these programs work by estimating the number of positive and negative real roots and narrowing down the search space.

Problem 1

Find positive and negative real roots for the given polynomial: f(x) = x3 − 6x2 + 11x − 6

Okay, lets begin

Possible positive real roots: 3, 1. There are no negative roots.
 

Explanation

  1. Positive roots:
    Sign changes: \(+x^3 → −6x^2 → +11x → −6\)
     → 3 sign changes, possible positive real roots can be: 3, 1
  2. Negative roots: Substitute \(f(−x) = (−x)^3 − 6(−x)^2 + 11(-x) - 6\)
    = \( -(x)^3 − 6(x)^2 - 11(x) - 6\)
    → No sign changes, so there are zero negative real roots

Well explained 👍

Problem 2

Apply Descartes' rule of signs to estimate positive and negative real roots of the polynomial f(x) = x4 + x3. - x -1

Okay, lets begin

 Positive real roots: 1 or 0, negative real roots: 3 or 1
 

Explanation

  1. Positive roots: sign changes: \(+x^4 → + x^3 → −x → −1\)
    → 1 sign change
  2. Negative roots: \(f(−x) = x^4 − x^3 + x − 1\)
    Sign changes: \(+x^4 → −x^3 → +x → −1\)
    → 3 sign changes

Well explained 👍

Problem 3

Determine the possible number of positive and negative real roots for the polynomial f(x) = x5 − 4x4 + 6x3 − 4x2 + x - 2 using Descartes' rule of signs

Okay, lets begin

Positive real roots: 4, 2, or 0. Negative real roots: 0
 

Explanation

  1. Positive real roots; 
    \(f(x) = +x^5, -4x^3, -4x^2, +x, -2\)
    Signs are +, -, +, -, +, -

    There are 5 sign changes, so the possible number of real roots is 5, 3, or 1.

  2. Negative real roots;
    \(f(-x) = (−x)^5 − 4(−x)^4 + 6(−x)^3 − 4(−x)^2 + (−x) − 2\\ =−x^5 − 4x^4 − 6x^3 − 4x^2 − x − 2\)

    There are no sign changes, so there are 0 negative real roots.

Well explained 👍

Problem 4

Use Descartes' rule to find the maximum number of positive and negative real roots for the polynomial f(x) = x3 + x2 + x +1

Okay, lets begin

There are zero positive roots and 3 or 1 negative roots.
 

Explanation

  1. Positive roots: All coefficients are positive, no changes in signs 
  2. Negative roots: \( f(−x) = −x^3 + x^2 − x + 1\)
    Sign changes:     \(−x^3 → +x^2 → −x → +1\)

    → 3 sign changes

Well explained 👍

Problem 5

Apply Descartes' rule of signs to the polynomial f(x) =x4 −5x2 + 4

Okay, lets begin

 Positive roots 
 

Explanation

  • Count the sign changes in f(x)
     
  1. Write the polynomial in standard form: \( f(x) = x^4 + 0x^3 - 5x^2 + 0x + 4\)
     
  2. Signs of coefficients are +, 0, -, 0, +
     
  3. There are a total of 2 sign changes, f(x) = 2
     
  4. By Descartes' rule of signs, this means that the number of positive real roots is either 2 or 0.
  • Count sign changes in f(-x)
  1. \(f(-x) = (-x)^2 + 4 = x^4 - 5x^2 + 4 = f(x)\)
     
  2. So f(-x) = f(x), which means the polynomial is even.
     
  3. There are 0 sign changes in f(-x)
     
  4. So, by Descartes' rule, the number of negative roots is zero.

Well explained 👍

FAQs on Descartes' rule of signs

1.How to help my child in learning the advantages and disadvantages of Descartes' rule of signs?

Encourage your child to make a chart of advantages and disadvantages of Descartes' rule of signs. Use colors and little characters to make the process fun and engaging. The chart will help your child to visually retract information.

The chart will include the following advantages and disadvantages.

Advantages

Disadvantages

Quick estimation of real roots.

Only gives possible numbers, not exact.

Helps eliminate impossible cases.

Gives no information on complex roots.

Easy to apply and only requires sign changes in coefficients.

The polynomial must be in descending order and like terms should be combined properly, otherwise the result will be wrong.

Useful in polynomial root isolation

It does not detect multiplicity

Supports early stability checks

Less informative and harder to narrow down for higher degree polynomials.

2.Why my child should learn Descartes' rule?

The rule explains the behavior of equations. It helps children to better understand the concepts of algebra and builds strong foundations. It allows them to find answers logically before using any tool.
 

3.Can my child find zeroes using Descartes' rule of signs?

 No, we can not directly find the exact roots using Descartes' rule.
 

4.Can my child use Descartes' rule if some coefficients are zero?

Yes, however, zero coefficients are skipped while counting sign changes. Sign changes are counted only for non-zero coefficients.
 

5.Will my child ever use Descartes' Rule in future?

Descartes Rule has wide applications. Children pursuing the field of engineering, physics, computer science, robotics, mathematics will use Descartes' rule in the future.
 

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.