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Math Help - Complex number question

  1. #1
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    Complex number question

    The question:
    Suppose both c and (1 + ic)^5 are real (c != 0). Show that c = \pm \sqrt{5 \pm 2\sqrt{5}}

    How do I solve this? I substituted the value of c into the complex number with the intent to find the magnitude and angle, but I'm not sure how to handle the \pm. Is it two different complex numbers? Am I attempting this correctly? Thanks.
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  2. #2
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    You can use the Binomial Theorem to expand \displaystyle (1 + ic)^5...

    \displaystyle (1 + ic)^5 = \sum_{r = 0}^n{{n\choose{r}}1^{n-r}(ic)^r}

    \displaystyle = 1 + 5ic + 10(ic)^2 + 10(ic)^3 + 5(ic)^4 + (ic)^5

    \displaystyle = 1 + 5ic -10c^2 - 10ic^3 + 5c^4 + ic^5

    \displaystyle = 1 - 10c^2 + 5c^4 + (5c - 10c^3 + c^5)i.


    Since you know that \displaystyle (1 + ic)^5 is real, that means the imaginary part is \displaystyle 0.

    So \displaystyle 5c - 10c^3 + c^5 = 0.

    You should now be able to solve for \displaystyle c.
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    Solving quartics isn't exactly my speciality. :/

    What's the best way to tackle c(5 - 10c^2 + c^4) = 0? Or can I just rewrite the given value of 'c' since I know that's going to be the solution anyway? :P
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  4. #4
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    Quote Originally Posted by Glitch View Post
    Solving quartics isn't exactly my speciality. :/

    What's the best way to tackle c(5 - 10c^2 + c^4) = 0? Or can I just rewrite the given value of 'c' since I know that's going to be the solution anyway? :P
    You already know that \displaystyle c \neq 0, so \displaystyle c^4 - 10c^2 + 5 = 0.

    If you let \displaystyle C = c^2 then the equation becomes \displaystyle C^2 - 10C + 5 = 0.

    I'm sure you can solve this for \displaystyle C, then use that to solve for \displaystyle c.
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    Oh dang. I forgot that trick. Sometimes I wonder if I'm a descendant of Homer J Simpson.

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    Quote Originally Posted by Glitch View Post
    Oh dang. I forgot that trick. Sometimes I wonder if I'm a descendant of Homer J Simpson.

    If you were Lisa or Maggie you'd be alright :P
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  7. #7
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    Quote Originally Posted by Prove It View Post
    If you were Lisa or Maggie you'd be alright :P
    See, I can't even mock myself correctly! :P
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    Quote Originally Posted by Glitch View Post
    See, I can't even mock myself correctly! :P
    Of course, that's assuming that Lisa and Maggie are really his kids... Now that I think about it, isn't Maggie's real dad Kang the alien? Hahahaha
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  9. #9
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    Quote Originally Posted by Prove It View Post
    Of course, that's assuming that Lisa and Maggie are really his kids... Now that I think about it, isn't Maggie's real dad Kang the alien? Hahahaha

    Kinky.
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  10. #10
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    Quote Originally Posted by Glitch View Post

    Kinky.
    Even moreso, considering the aliens are mormons >< HAHAHA
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