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Math Help - Roots of 3rd degree polynomials and Characteristic Polynomials

  1. #1
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    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!!!
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  2. #2
    Super Member TheChaz's Avatar
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    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...
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  3. #3
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    o.o No. Wolframalpha is not a solution, you can't use it in an exam.
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    Super Member TheChaz's Avatar
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    Quote Originally Posted by veileen View Post
    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
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    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!
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    Super Member TheChaz's Avatar
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    Quote Originally Posted by lisaalg View Post
    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.
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  7. #7
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    "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.
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  8. #8
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    Quote Originally Posted by lisaalg View Post
    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!!

    In your example we have

    \begin{vmatrix} x-3 & 0 & 7 \\ <br />
0 & x-5 & 0 \\<br />
-7 & 0 & x-3<br />
\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!

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

    Now this is already partially factored for use


    (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

    x=5,x=3\pm7i
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