Finding eigenvalues to use in Cayley-Hamilton theorem problem

[phosgene]

Homework Statement

Let C =

2,0,-2
1,1,2
-1,-1,-1

Use the Cayley-Hamilton theorem to compute C^3.

Homework Equations

Cayley-Hamilton theorem says that every square matrix satisfies its own characteristic equation.

[itex]C^3=PD^3P^{-1}[/itex]

where P is the matrix formed from linearly independant eigenvectors of C and D is the diagonal matrix formed from the eigenvalues of C.

The Attempt at a Solution

I get the characteristic equation of C is

[itex]\lambda^3 - 2\lambda^2 - \lambda - 2 = 0 [/itex]

I get stuck because I can't factorise this and get the eigenvalues to proceed. Is there some trick to factorising cubics like this?

 

[hedipaldi]

Attached

 

[phosgene]

Thanks, I was just going back over the lecture notes and realized that I was absurdly confused in that section (I'm embarassed I even asked this question!)...anyway, I get it now, thanks for that :)