# Knowledge Base

## Solution 10349: Calculation Method Used by the Polynomial Root Finder on the TI-85 and TI-86.

### What calculation method is used by the polynomial root finder on the TI-85 and TI-86?

The TI-85 polynomial root finder makes use of the built-in eigenvalue solver. The coefficients of the polynomial are used to create what is called a "companion matrix". Basically, the coefficients appear across a row or column (there are variations) and the rest of the matrix is all zeros except for a pattern of ones just off the main diagonal.

The property of this matrix is that the characteristic polynomial of the matrix will be the same as the polynomial used to create the companion matrix. Since the eigenvalues of a matrix are the roots of its characteristic polynomial, the eigenvalue solver gives the desired answers.

This has the advantage of giving real and complex roots in one straightforward operation even for high order polynomials. This does not imply that a closed form solution for roots of polynomials of order 5 and greater has been found. The QR method for computing the eigenvalues is an iterative technique and is not guaranteed to converge to a solution.