Descartes' Rule of Signs

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.  

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

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 \(f(x) = 2x^4 - 3x^3 - 5x^2 + 9x - 4 \):

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.

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.

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.