What would be the easiest way to get the charateristic polynomial roots?!
What I'm actually doing right now is...
Given a matrix $A$, I calculate $\det(A-\lambda I) = 0$ with Laplace Expansion, and then, expand all terms to get usually a third degree equation (depends on the size of my matrix $A$)... Then I try to find one value for $\lambda$ that is root of that expression... Then I make a polynomial division, with the first expression (third degree equation) divided by $(x-a)$ when $a$ is the root that I've found. Then I get a second degree equation and use bhaskara method to get the other two roots. That method works, but it takes a long time and I can have many algebraic mistakes during the proccess, and if my first equation has degree $>3$ I'll be lost.
What is the proccess that I should adopt after the Laplace Expansion at $\det(A-\lambda I) = 0$? How can I simplify what I am doing?!