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Math Help - Polynomial iteration

  1. #1
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    Polynomial iteration

    Hello,

    given is a polynomial P(x)=x^2+4x+2. Find all solutions to the equation P^n(x)=0, where P^n(x)=\underbrace{ P(P(...P }_{n}(x)...)).
    I have managed to work out that the solutions are of the form \pm\sqrt[n+1]{2}-2, but to prove it is quite a different matter. I tried induction but soon wound up in a pile of useless symbols. I depend on you with this one!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by atreyyu View Post
    Hello,

    given is a polynomial P(x)=x^2+4x+2. Find all solutions to the equation P^n(x)=0, where P^n(x)=\underbrace{ P(P(...P }_{n}(x)...)).
    I have managed to work out that the solutions are of the form \pm\sqrt[n+1]{2}-2, but to prove it is quite a different matter. I tried induction but soon wound up in a pile of useless symbols. I depend on you with this one!
    Let \xi_n=\pm\sqrt[n]{2}-2. Clearly P^1(\xi_1)=0. Now, suppose that P^n(\xi_n)=0. Then, P^{n+1}(\xi_{n+1}) =\left(P^n(\xi_{n+1})\right)^2+4P(\xi_{n+1})+2=-\left(P^n(\xi_{n+1})+\sqrt{2}-2\right)\left(P^{n}(\xi_{n+1})-\sqrt{2}+2\right). That should make it more obvious how to proceed
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  3. #3
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    Hm... I'm trying to find a relation between \xi_{n+1} and \xi_n so that I can show that the last brackets are equal zero... is that the right direction for me to go?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by atreyyu View Post
    Hm... I'm trying to find a relation between \xi_{n+1} and \xi_n so that I can show that the last brackets are equal zero... is that the right direction for me to go?
    For sure!
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  5. #5
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    All I'm getting is \zeta_{n+1} = \sqrt[2^{n+1}]{1/2} \times \zeta_n -2+ \sqrt[2^{n+1}]{2^{2^{n+1}-1}}. Even if I plug it into one of these brackets, I can't do anything about the P_n(...). I'm stuck :/
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