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I'm doing a course in linear algebra at the moment, and whenever I need to find the eigenvalues of a 3x3 matrix, I'm faced with the issue that I don't know a general method for solving a cubic equation, or I don't remember enough from high school algebra to be able to solve any given cubic equation.
For instance, if I have the characteristic equation
then I can't factor it easily because 19 is a prime number, but a computer was able to somehow find it's solutions for me.
Is there a general, algorithmic method for solving any given cubic equation without factorization? If not, how would you solve an equation like the above?
And is there a similar kind of general method for solving quartic or higher degree polynomials?