Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Use a CAS or – Amzoti Jul 22 '13 at 0:39

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:



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 $x=1,2,3$.

share|cite|improve this answer
This is a nice way to find the rational root. – eccstartup Jul 22 '13 at 1:03


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

share|cite|improve this answer

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.

share|cite|improve this answer
Okay, I've got it now! Thanks! – briteId Jul 22 '13 at 0:43
@Cee If the answer helped you, don't forget to select it (the check mark under the voting arrows). – Ataraxia Jul 22 '13 at 0:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.