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Complex numbers and roots

  1. Jan 30, 2013 #1
    1. The problem statement, all variables and given/known data
    Let z be a complex number satisfying the equation ##z^3-(3+i)z+m+2i=0##, where mεR. Suppose the equation has a real root, then find the value of m.


    2. Relevant equations



    3. The attempt at a solution
    The equation has one real root which means that the other two roots are complex and are conjugates of each other.
    Let ##\alpha, \beta, \gamma## be the three roots. The coefficient of z^2 is zero.
    Hence ##\alpha+\beta+\gamma=0##. Let ##\gamma## be the real root and the other two are complex. The sum of the complex roots is zero. From here, i get ##\gamma=0##.
    Product of roots, ##\alpha \beta \gamma=m+2i=0##. This gives me ##m=-2i## which is incorrect as m is a real number.

    Any help is appreciated. Thanks!
     
  2. jcsd
  3. Jan 30, 2013 #2

    mfb

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    Staff: Mentor

    Why? Their imaginary parts cancel, but the real parts do not have to.

    Your approach is way too complicated. If the equation has a real root, there is a real z which satisfies the equation. What is the imaginary part of the left side? It has to be zero...
    This allows to calculate z and m.
     
  4. Jan 30, 2013 #3
    Oops, sorry about that.

    Do you mean I have to substitute z=x+iy and compare the real and imaginary parts?
     
  5. Jan 30, 2013 #4

    mfb

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    Staff: Mentor

    What is the imaginary part of z^3 if z is real? What is the imaginary part of (3+i)z?
    m is real, and the imaginary part of 2i is obvious.
     
  6. Jan 30, 2013 #5
    Thanks a lot mfb! :smile:
     
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