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Polynomial roots



r = roots(p) returns the roots of the polynomial represented by the coefficients in p as a column vector r. Input p is a vector containing n+1 polynomial coefficients, starting with the coefficient of xn. For example, p = [3 2 -2] represents the polynomial 3x2+2x2. A coefficient of 0 indicates an intermediate power that is not present in the equation.

The roots function solves polynomial equations of the form p1xn+...+pnx+pn+1=0. Polynomial equations contain a single variable with nonnegative exponents.


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Solve the equation 3x2-2x-4=0.

Create a vector to represent the polynomial, then find the roots.

p = [3 -2 -4];
r = roots(p)
r = 2×1


Solve the equation x4-1=0.

Create a vector to represent the polynomial, then find the roots.

p = [1 0 0 0 -1];
r = roots(p)
r = 4×1 complex

  -1.0000 + 0.0000i
   0.0000 + 1.0000i
   0.0000 - 1.0000i
   1.0000 + 0.0000i

Input Arguments

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Polynomial coefficients, specified as a vector. For example, the vector [1 0 1] represents the polynomial x2+1, and the vector [3.13 -2.21 5.99] represents the polynomial 3.13x22.21x+5.99.

For more information, see Create and Evaluate Polynomials.

Data Types: single | double
Complex Number Support: Yes


  • Use the poly function to obtain a polynomial from its roots: p = poly(r). The poly function is the inverse of the roots function.

  • Use the fzero function to find the roots of nonlinear equations. While the roots function works only with polynomials, the fzero function is more broadly applicable to different types of equations.


The roots function considers p to be a vector with n+1 elements representing the nth degree characteristic polynomial of an n-by-n matrix, A. The roots of the polynomial are calculated by computing the eigenvalues of the companion matrix, A.

A = diag(ones(n-1,1),-1);
A(1,:) = -p(2:n+1)./p(1);
r = eig(A)

The results produced are the exact eigenvalues of a matrix within roundoff error of the companion matrix, A. However, this does not mean that they are the exact roots of a polynomial whose coefficients are within roundoff error of those in p.

Extended Capabilities

Version History

Introduced before R2006a