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Complex Number equations from roots

  1. Feb 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Determine the only real values a, b, c, and d such that the equation:
    z4+az3+bz2+cz+d = 0​
    has both z1 and z2 as roots.

    z1 = 3 + j
    z2 = -5 + 5j

    2. Relevant equations
    z = x + yj.
    z = |z|ej[itex]\theta[/itex]

    3. The attempt at a solution
    I am not sure where to begin. I can convert between Cartesian coordinates and Euler's formula but don't know where to go from there.

    Any help would be greatly appreciated!
  2. jcsd
  3. Feb 3, 2012 #2


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    Hi DmytriE! :smile:

    Hint: if 3 + j is a root (of a real-coefficient equation), can you spot another root? :wink:
  4. Feb 3, 2012 #3
    Indeed! Now, I do. Thanks for the hint!
    Last edited: Feb 3, 2012
  5. Feb 3, 2012 #4
    I have calculated the different combination of roots but how do I know which variable (a,b,c,d) the values go to? When I multiplied z1, z1*, z2, and z2* I got a value of 500. Would that go into d?

    It seems logical when I compare it to a second degree polynomial...
  6. Feb 3, 2012 #5


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    But why not do it all in one go? …

    Find (z - z1)(z - z1*), then (z - z2)(z - z2*), then multiply them. :wink:
  7. Feb 3, 2012 #6
    Great! Thank you again. I keep forgetting that z is a variable so I was strictly looking for the number rather than the equation with the coefficients.
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