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Math Help - how to prove z0 is a root of the polynomial

  1. #1
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    how to prove z0 is a root of the polynomial

    How can I prove that z_0 and \overline{z_{0}} is a root of the polynomial
    a_{n}z^n + a_{(n-1)} z^{(n-1)} +....+a_{1} z + a_{0} = 0

    Using (\bar{z})^k = \overline{z^k}

    I have been able to prove (\bar{z})^k = \overline{z^k} but I dont get how to use that to show that z_0 and \overline{z_{0}} are roots of the polynomial.

    Thanks in advance for your help!
    Last edited by flametag3; November 1st 2012 at 10:44 AM. Reason: Incorrect formula
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  2. #2
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    Re: how to prove z0 is a root of the polynomial

    Hey flametag3.

    Is this asking to show that if z0 is a root then z0_bar is a root or is it asking that if the condition (z_bar)^k = (z^k)_bar, then that particular z is a root for when that is true?
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  3. #3
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    Re: how to prove z0 is a root of the polynomial

    Hey Chiro,

    thanks for your reply. Its asking to show that if z0 is a root then z0_bar is a root as well.
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  4. #4
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    Re: how to prove z0 is a root of the polynomial

    You'll have to show that if z gives 0 then z_bar also gives 0, and one to show that given what you have is to show that the equation with z_bar is zero if the one with z is zero and you will need to apply this term by term in your polynomial to show that both equal zero and thus both are roots of the polynomial if they have a specific f(z) form (note f(z) doesn't change but z = z or z = z_bar both need to give f(z) = 0 to both be roots).
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