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Thread: Complex numbers

  1. #1
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    Complex numbers

    Please help me answer both of these questions. Thank you
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by norivea
    Please help me answer both of these questions. Thank you
    1a) Find the real and complex roots of:

    $\displaystyle
    (z-3)(z^2-5z+8)=0
    $

    This has three roots, one real corresponding to the first factor; so $\displaystyle x=3$ is a root.

    The other two roots are the roots of the second factor $\displaystyle z^2-5z+8=0$, these may
    be found using the quadratic formula, which gives in this case:

    $\displaystyle
    z=\frac{5}{2}\pm \frac{\sqrt{7}}{2}i
    $

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by norivea
    Please help me answer both of these questions. Thank you
    1b) Find the real and complex roots of:

    $\displaystyle
    z^3-10z^2+34z-40
    $

    given that $\displaystyle 3-i$ is a root.

    A cubic with real coefficients always has at least one real root, and
    complex roots occur in conjugate pairs. So both $\displaystyle 3-i$ and
    $\displaystyle 3+i$ are roots. Also:

    $\displaystyle
    z^3-10z^2+34z-40=(z-(3-i))(z-(3+i))(z-a)
    $

    where $\displaystyle a$ is real, and so easily found to be $\displaystyle 4$,
    so the remaining root is $\displaystyle z=4$.

    RonL
    Last edited by CaptainBlack; Jun 16th 2006 at 04:10 AM.
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