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Math Help - Need Assistance with proof of the existence of conjugate complex roots

  1. #1
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    Need Assistance with proof of the existence of conjugate complex roots

    Prove:
    given that f(x) = Anx^n + An-1X^n-1+ ....A1X + A0 , and An does not = 0
    => if f(x) has a root of the form (A+Bi),then it must have a root of the form
    (A-Bi). (complex roots)

    All I can figure out so far is to utilize the Conjugate Roots theorem but our professor gave us a hint that we should use the Remainder theorem? or the discriminate? I'm not quite sure but basically I need to Prove why the conjugate exists in the polynomial.

    Regards, Chris
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  2. #2
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    The way you have stated the question makes it false.
    Consider z^2  + \left( {2 - i} \right)z + 1. There are two complex roots that are not conjugates.

    The usual statement includes A_n  \in \Re . Now it is a theorem.
    Here is an outline of a way to prove it.
    If w is a root of f(z) then Re(f(w)) = Re(f(\bar w)) and Im(f(w)) =  - Im(f(\bar w)).
    Can you show that? If so it follows that {\bar w} is also a root.
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  3. #3
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    Unhappy

    Appreciate the fast reply, But I'm not quite understanding your notations.
    Re(f(w))...Im(f(w)), What does the Re, Im, W-bar mean?

    Also, I'm assuming the coefficients are real too.

    Need more of an step by step explanation over the computer.
    Thanks for your time and effort.

    Regards, Chris
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