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Complex numbers proof

  1. Nov 1, 2015 #1
    1. The problem statement, all variables and given/known data

    I have two complex numbers that are non real, k and z. K and z are going to be complex conjugates if and only if the product (x-k)(x-z) is a polynomial with real coefficients.

    Here is my answer :

    k=a+bi

    z=c+di

    (x-k)(x-z) = x^2 -(k+z)x+kz

    2. Relevant equations


    3. The attempt at a solution
    I was able to prove that a=c and d=-b (I have proven they're conjugatCes)

    But because this is a if and only if, I must prove that if they're conjugates, then the coefficients are real. How do I do that ?
     
  2. jcsd
  3. Nov 1, 2015 #2

    BvU

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    So k = a + bi and z = a - bi. Now write out (x-k)(x-z) and see if there's anything complex left over or not !
     
  4. Nov 1, 2015 #3
    To be honest, I did think about doing that but I was lazy and didn't try it and went just to ask this question... Thank you !
     
  5. Nov 1, 2015 #4

    BvU

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    Being lazy is often a good quality for finding an economic way out of a problem :smile:
     
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