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Math Help - Factorising polynomial over complex numbers

  1. #1
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    Factorising polynomial over complex numbers

    The question:
    Factorise the polynomial z^4 + 2z^2 - 3 over the complex numbers.

    My attempt:
    Let u = z^2

    u^2 + 2u -3 = 0

    \frac{-2 \pm \sqrt{4 - 4(1)(-3)}}{2}

    u = 1, -3

    (u - 1)(u + 3) = 0
    (z^2 - 1)(z^2 + 3) = 0

    What do I do from here? Do I take z^2 = 1 (for example) then apply De'Movres theorem to find the roots of unity? Or is there a better way? Thanks.
    Last edited by Glitch; March 19th 2011 at 05:50 PM. Reason: Forgot to add square in Q
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Glitch View Post
    The question:
    Factorise the polynomial z^4 + 2z - 3 over the complex numbers.

    My attempt:
    Let u = z^2

    u^2 + 2u -3 = 0

    \frac{-2 \pm \sqrt{4 - 4(1)(-3)}}{2}

    u = 1, -3

    (u - 1)(u + 3) = 0
    (z^2 - 1)(z^2 + 3) = 0

    What do I do from here? Do I take z^2 = 1 (for example) then apply De'Movres theorem to find the roots of unity? Or is there a better way? Thanks.
    If u=z^2

    2u != 2z


    Suppose you got:

    (z^2 - 1)(z^2 + 3)

    (z^2 - 1)(z^2 + 3) =(z-1)(z+1)(z-i*sqrt(3))(z+i*sqrt(3))
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  3. #3
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    Quote Originally Posted by Also sprach Zarathustra View Post
    If u=z^2

    2u != 2z
    Sorry, forgot to add the square to the Q.
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  4. #4
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    \displaystyle u^2 + 2u - 3 = (u - 1)(u + 3)

    \displaystyle = (z^2 - 1)(z^2 + 3)

    \displaystyle = (z - 1)(z + 1)[z^2 - (i\sqrt{3})^2]

    \displaystyle = (z - 1)(z + 1)(z - i\sqrt{3})(z + i\sqrt{3}).

    The solutions should now be obvious.
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  5. #5
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    Because those are roots of purely real numbers, you don't really need to use DeMoivre's formula.
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  6. #6
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    Quote Originally Posted by HallsofIvy View Post
    Because those are roots of purely real numbers, you don't really need to use DeMoivre's formula.
    Now that I think about it, that's pretty obvious. I hope my brain is functioning better tomorrow for my exam. >_<

    Thanks again guys!
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