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Math Help - complex numbers question

  1. #1
    Senior Member
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    complex numbers question

    Hi,
    firstly, is this the appropriate place to post questions related to complex numbers? I couldn't seem to find a better place.. so move this if its in the wrong area.
    EDIT: i just realized this would probably be better in the Advanced algebra section...

    ok, my question is this:
    Given that z = 2 + 3i is a solution to the equation z^3 - 6z^2 + 21z -26 = 0
    a) write down another complex solution of the equation.
    I wrote 2 - 3i (the conjugate). is that correct?
    b) Hence find a quadratic factor of the left hand side of the equation
    c) Find the third solution.

    I'm not sure how to do the rest of those...
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by scorpion007 View Post
    Hi,
    firstly, is this the appropriate place to post questions related to complex numbers? I couldn't seem to find a better place.. so move this if its in the wrong area.
    EDIT: i just realized this would probably be better in the Advanced algebra section...

    ok, my question is this:
    Given that z = 2 + 3i is a solution to the equation z^3 - 6z^2 + 21z -26 = 0
    a) write down another complex solution of the equation.
    I wrote 2 - 3i (the conjugate). is that correct?
    The complex roots of polynomial equations with real coefficients always
    occur in conjugate pairs, so yes this is right.

    b) Hence find a quadratic factor of the left hand side of the equation
    You have two factors of your cubic (z-(2+3i)), and (z-(2-3i)). If you multiply
    these together you will get a quadratic factor of the cubic with real coefficients.
    This is the quadratic factor you are looking for here.

    c) Find the third solution.
    Use polynomial long division of your cubic by your quadratic factor to find
    the remaining linear factor (it can also be done by inspection). This will
    be a real linear factor and from it you can find the third root of the
    equation.

    RonL
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  3. #3
    Senior Member
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    thank you very much, i worked it out
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