The polynomial given by the determinant of:
\begin{equation} det(A-\lambda I) \end{equation}
for some Linear Map A. Solutions for \lambda are the eigenvalues. This is because something is an eigenvalue IFF (A-\lambda I)v = 0 for some \lambda, v, so we need (A-\lambda I) to be singular. Characteristic polynomial of a 2x2 matrix is given by \lambda^{2}-tr(A)\lambda + det(A).