Polynomial iteration

**atreyyu** · Apr 1st 2010, 03:14 PM

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!

**Drexel28** · Apr 1st 2010, 05:01 PM

Originally Posted by atreyyu

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

**atreyyu** · Apr 2nd 2010, 07:08 AM

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?

**Drexel28** · Apr 2nd 2010, 10:10 AM

Originally Posted by atreyyu

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!

**atreyyu** · Apr 2nd 2010, 01:31 PM

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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