MATLAB - Polynomial Division

MATLAB - Modern Polynomial Division

Polynomial division is a core algebraic operation used to simplify fractions of polynomials, find roots, and analyze transfer functions in control systems. In MATLAB, the primary function for numerical polynomial division is deconv, which stands for deconvolution—the inverse operation of polynomial multiplication (conv).

Using the deconv Function

The deconv function divides a numerator polynomial by a denominator polynomial, returning the quotient and the remainder.

Syntax

[quotient, remainder] = deconv(numerator_poly, denominator_poly)

Where numerator_poly and denominator_poly are row vectors of coefficients in descending order of power. The output quotient and remainder are also coefficient vectors.

Example 1: Basic Division

Let’s divide P(x) = 6x³ + 5x² + 4x + 3 by Q(x) = 2x + 1.

%% Define the coefficient vectors for the numerator and denominator
P = [6 5 4 3];  % Represents 6x³ + 5x² + 4x + 3
Q = [2 1];    % Represents 2x + 1

%% Perform the polynomial division
[quotient, remainder] = deconv(P, Q);

%% Display the results
disp('Quotient coefficients:');
disp(quotient); % Expected: [3 1 1.5]

disp('Remainder coefficients:');
disp(remainder); % Expected: [0 0 0 1.5]

fprintf('Result: (%sx^2 + %sx + %s) with a remainder of %s\n', ...
    num2str(quotient(1)), num2str(quotient(2)), num2str(quotient(3)), ...
    num2str(remainder(end)));
% The remainder vector is padded with leading zeros to match the length of P.

Verification and Numerical Precision

You can verify the result of the division using the fundamental relationship: Numerator = Quotient * Denominator + Remainder. In MATLAB, this is done using the conv and + operators.

Pro Tip: When working with floating-point numbers, the remainder might not be exactly zero due to precision limits. It’s good practice to check if it’s smaller than a small tolerance (e.g., 1e-12).

Example 2: Exact Division and Verification

Let’s divide P(x) = x⁴ + 4x³ + 6x² + 4x + 1 (which is (x+1)⁴) by Q(x) = x + 1. The result should be (x+1)³ with a zero remainder.

%% Define polynomials for exact division
P = [1 4 6 4 1]; % (x+1)^4
Q = [1 1];       % (x+1)

[quotient, remainder] = deconv(P, Q);

disp('Quotient (x+1)^3 coefficients:');
disp(quotient); % Expected: [1 3 3 1]

disp('Remainder coefficients:');
disp(remainder); % Expected: [0 0 0 0 0]

%% Verification Step
% Reconstruct the original polynomial P
reconstructed_P = conv(quotient, Q) + remainder;

% Check if the reconstruction is close to the original
is_verified = all(abs(P - reconstructed_P) < 1e-12);

if is_verified
    disp('Verification successful: P = conv(Q, Quotient) + Remainder');
else
    disp('Verification failed.');
end

Modern Approach: Symbolic Math Toolbox

For analytical work, where precision is paramount and you want to work with expressions rather than just coefficients, the Symbolic Math Toolbox is the superior choice. It performs exact rational arithmetic.

% This requires the Symbolic Math Toolbox
try
    %% Define symbolic variable and polynomials
    syms x;
    sym_P = x^4 + 3*x^2 + 1; % Numerator
    sym_Q = x^2 + x + 1;     % Denominator

    %% The `quorem` function performs symbolic division
    [sym_quotient, sym_remainder] = quorem(sym_P, sym_Q, x);

    %% Display the symbolic results
    fprintf('Symbolic Quotient: %s\n', string(sym_quotient));
    fprintf('Symbolic Remainder: %s\n', string(sym_remainder));
catch ME
    if strcmp(ME.identifier, 'MATLAB:UndefinedFunction')
        disp('Symbolic Math Toolbox is not installed. Skipping this example.');
    else
        rethrow(ME);
    end
end

Practical Application: System Analysis

In control theory, a transfer function is a ratio of two polynomials, G(s) = N(s) / D(s). If N(s) has a higher degree than D(s) (an improper transfer function), deconv can be used to separate it into a proper transfer function and a feed-forward term, which is crucial for analysis and simulation.