# Roots of 3rd degree polynomials and Characteristic Polynomials

• Apr 7th 2011, 11:54 AM
lisaalg
Roots of 3rd degree polynomials and Characteristic Polynomials
Hi there,

I am in desperate need of some algebra help! I'm working with matrices and diagonalization. One step of this is finding the C(x) (characteristic polynomial) which entails find the determinant of a matrix with x's along its diagonal. So this is the matrix I have:

x-3 0 7
0 x-5 0
-7 0 x-3

For the determinant I get: x^3 - 11x^2 +88x -290

To continue, I need to find the roots of this, and I have no clue how to do that. I put the matrix in here so that if I made any mistakes with the determinant you can let me know. Your help would be much appreciated! I have an exam tomorrow and I'm quite a bit behind with studying... :( Just please, if you can, answer in really simple terms, because I am pretty new at this.

Thank you!!!
• Apr 7th 2011, 11:59 AM
TheChaz
Solving polynomial equations happens in a few ways. The easiest is technology(!) - wolframalpha.com just spit out the three roots for me.
x = 5 is the only real one.

Other methods would be factoring, using the (potential) rational root theorem with synthetic division, numerical methods...
• Apr 7th 2011, 12:05 PM
veileen
o.o No. Wolframalpha is not a solution, you can't use it in an exam.
• Apr 7th 2011, 12:08 PM
TheChaz
Quote:

Originally Posted by veileen
o.o No. Wolframalpha is not a solution, you can't use it in an exam.

Non sequitor. Also, you can't use MHForum on an exam!

For people doing math beyond what school requires of them, solutions come in many forms and fashions ;)
• Apr 7th 2011, 12:17 PM
lisaalg
I tried using synthetc division, but I'm having a hard time figuring out what to divide by. Also what do I do if I get a remainder? The format of the answer should be sth like this:
(x +/- a) (x +/- b) (x +/- c) so that I can work with it.

Could you maybe show me how to do it in this case?

P.S.: I know I can't use MHF on the exam, but I could learn from it and use that. :) If a tech program just gives me the answer, that I def can't use!
• Apr 7th 2011, 12:25 PM
TheChaz
Quote:

Originally Posted by lisaalg
I tried using synthetc division, but I'm having a hard time figuring out what to divide by. Also what do I do if I get a remainder? The format of the answer should be sth like this:
(x +/- a) (x +/- b) (x +/- c) so that I can work with it.

Could you maybe show me how to do it in this case?

P.S.: I know I can't use MHF on the exam, but I could learn from it and use that. :) If a tech program just gives me the answer, that I def can't use!

re: p.s. I was talking to the other guy!

How to know what to use in sythetic division? I mentioned it already, but here's a link:
Rational root theorem - Wikipedia, the free encyclopedia

If you get a remainder, then whatever you synthetically divided by is not a root.

Furthermore, if x = a is a root, then (x - a) is a factor.
• Apr 7th 2011, 12:38 PM
veileen
"p.s. I was talking to the other guy!" ._. You're the second person that think i'm a "he".

"Also, you can't use MHForum on an exam!" You are right, but you can learn here and don't need help on an exam. Using Wolframalpha you learn nothing.

"For people doing math beyond what school requires of them, solutions come in many forms and fashions." I don't think this problem is beyond what school requires.
• Apr 7th 2011, 12:38 PM
TheEmptySet
Quote:

Originally Posted by lisaalg
Hi there,

I am in desperate need of some algebra help! I'm working with matrices and diagonalization. One step of this is finding the C(x) (characteristic polynomial) which entails find the determinant of a matrix with x's along its diagonal. So this is the matrix I have:

x-3 0 7
0 x-5 0
-7 0 x-3

For the determinant I get: x^3 - 11x^2 +88x -290

To continue, I need to find the roots of this, and I have no clue how to do that. I put the matrix in here so that if I made any mistakes with the determinant you can let me know. Your help would be much appreciated! I have an exam tomorrow and I'm quite a bit behind with studying... :( Just please, if you can, answer in really simple terms, because I am pretty new at this.

Thank you!!!

You can save yourself a TON of work if you use the properties of matrices!!

$\displaystyle \begin{vmatrix} x-3 & 0 & 7 \\ 0 & x-5 & 0 \\ -7 & 0 & x-3 \end{vmatrix}$

Since we are taking a determinant expand across the middle row(you are free to choose) we will only have 1 term in our co-factor expansion because the other were multiplied by 0!

$\displaystyle (x-5)\begin{vmatrix} x-3 & 7 \\ -7 & x-3\end{vmatrix}$

Now this is already partially factored for use

$\displaystyle (x-5)\begin{vmatrix} x-3 & 7 \\ -7 & x-3\end{vmatrix}=(x-5)[(x-3)^2+49]$

From here is it easy to get the eigenvaules

$\displaystyle x=5,x=3\pm7i$