Find the eigenvalues of the matrices.

Question

The characteristic equations for the two matrices are:

$x^3-8x-7=0$ and $x^3-6x^2+11x-6=0$

I know that in order to find the eigenvalues, I need to factor these two equations out. I'm just having a brain freeze on how to factor cubic polynomials. Can anyone refresh my memory on solving these?

Answer 1

Hint

Notice that $-1$ is a root for the first polynomial and $1$ for the second.

Answer 2

To expand on Sami Ben Romdhane's answer, typically the strategy for factoring homework polynomials is to guess integer values near zero which may be roots.

Once you've got some root $r$, you can factor the polynomial by dividing it by $(x - r)$ (using long division).

There does exist an analogue of the quadratic equation, but it is roughly a page long and I've never seen it used.

Answer 3

A very useful tool in factoring $n$ degree polynomials is the rational root theorem. It states that all rational roots of a polynomial are a ratio between individual prime factors of the constant term, and the leading coefficient:

$$x^3-6x^2+11x-6=0$$

$$x=\pm\frac{3,2,1}{1}$$

So assuming there are rational roots, they are either $x=\pm3$, $x=\pm2$, or $x=\pm1$. By trial and error (plugging them in to see if any of them evaluate the polynomial to $0$), you can find that the roots are $x=1,2,3$.