Factorize a third degree polynomial I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but when it comes to finding the roots, I have a third degree polynomial and I don't know how to factorize it to get the eigenvalues associated. All the solutions on the internet and here about factorizing third degree polynomial are about specific case/obvious solutions and does not give a clear method like the method to factorize a second degree polynomial with steps. Could you please provide me a method to find roots in every third degree polynomial? If not this is the polynomial I found that I need to factorize : $X^3 - 3X - 2$ Thank you for taking your time to read my problem.

Try to "guess" some rational root $\;\cfrac rs\;$ , which by the Rational Root Theorem must fulfill $\;r\,\mid\,-2\;,\;\;s\,\mid\,1\;$ , and indeed $\;2\;$ is a root, so divide by $\;x-2\;$ :

$$x^3-3x-2=(x-2)(x^2+2x+1)=(x-2)(x+1)^2$$

and you have one simple root and one double one.

If there is no rational root then the task is much, but really much harder _in the general case_