I'm currently trying to solve a problem which asks if a 3×3 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.
Best Answer
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