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Solving a complex equation

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve (z+1)^5 = z^5


    2. Relevant equations

    None

    3. The attempt at a solution

    z^5 + 5z^4 + 10z^3 + 10z^2 + 5z + 1 = z^5
    5z^4 + 10z^3 + 10z^2 + 5z + 1 = 0
    5z^3(z + 2) + 5z(2z + 1) = -1

    I'm not quite sure how to go about solving this. Expanding, canceling terms, and then factoring doesn't get me anywhere.
     
  2. jcsd
  3. Sep 20, 2012 #2

    jbunniii

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    It's clear that there are no real roots: consider the graphs of (x+1)^5 and x^5 for real x. Are you looking for an exact solution or a numerical approximation? For an exact solution, you'll have to solve a quartic equation, which can be done but it's fairly ugly.
     
  4. Sep 20, 2012 #3

    SammyS

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    Play with it.

    5z4 + 10z3 + 10z2 + 5z + 1
    =5(z4 + 2z3 + 2z2 + z) + 1

    =5(z4 + 2z3 + z2 + z2 + z) + 1

    =5( (z2 + z)2 + (z2 + z) ) + 1

    =5 (z2 + z)2 + 5 (z2 + z) + 1 ​

    Let u = z2 + z .

    You have a quadratic equation in u .

    Solve for u, then solve u = z2 + z for z.

    Added in Edit:

    See Dick's method in the next post. Sweet!
     
    Last edited: Sep 20, 2012
  5. Sep 20, 2012 #4

    Dick

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    Or just write your equation as [itex](\frac{z+1}{z})^5=1[/itex]. That tells you 1+1/z is a fifth root of unity. It's pretty easy from there.
     
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