Not to mention that when your matrix is 5x5 or more, there aren't even general solutions for roots if for some reason you're still insisting on the matrix->polynomial->eigenvalues route.
It's not that eigenvalues per se are useless--they're plainly not--but that no one finds eigenvalues in practice by computing the characteristic polynomial and solving for its roots. Unfortunately, that computation is often found in HW and exams in US undergraduate linear algebra courses.