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This question already has an answer here:

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

$$-\lambda^3+8\lambda^2-19\lambda+12=0$$

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?

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marked as duplicate by Ross Millikan algebra-precalculus Oct 4 '14 at 14:40

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Hint: Use Rational roots theorem.

The R.R. Theorem says the rational roots of $-\lambda^3 + 8\lambda^2 -19\lambda + 12$ have to be of the form $$ \pm \frac{a}{1} $$ where $a$ is a factor of $12$

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For the cubic equations you can use (even if is really complicated) the Cardano Formula.

There are results in Agebra that show there are no such closed formulas for higher degree equations. This depends on the non-solvability property of permutations groups $S_n$ for $n>4$

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