Thread: Polynomial iteration

Hello,

given is a polynomial $\displaystyle P(x)=x^2+4x+2$. Find all solutions to the equation $\displaystyle P^n(x)=0$, where $\displaystyle P^n(x)=\underbrace{ P(P(...P }_{n}(x)...))$.
I have managed to work out that the solutions are of the form $\displaystyle \pm\sqrt[n+1]{2}-2$, 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!

Originally Posted by atreyyu

Hello,

given is a polynomial $\displaystyle P(x)=x^2+4x+2$. Find all solutions to the equation $\displaystyle P^n(x)=0$, where $\displaystyle P^n(x)=\underbrace{ P(P(...P }_{n}(x)...))$.
I have managed to work out that the solutions are of the form $\displaystyle \pm\sqrt[n+1]{2}-2$, 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 $\displaystyle \xi_n=\pm\sqrt[n]{2}-2$. Clearly $\displaystyle P^1(\xi_1)=0$. Now, suppose that $\displaystyle P^n(\xi_n)=0$. Then, $\displaystyle P^{n+1}(\xi_{n+1})$$\displaystyle =\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

Hm... I'm trying to find a relation between $\displaystyle \xi_{n+1}$ and $\displaystyle \xi_n$ so that I can show that the last brackets are equal zero... is that the right direction for me to go?

Originally Posted by atreyyu

Hm... I'm trying to find a relation between $\displaystyle \xi_{n+1}$ and $\displaystyle \xi_n$ so that I can show that the last brackets are equal zero... is that the right direction for me to go?

For sure!

All I'm getting is $\displaystyle \zeta_{n+1} = \sqrt[2^{n+1}]{1/2} \times \zeta_n -2+ \sqrt[2^{n+1}]{2^{2^{n+1}-1}}.$ Even if I plug it into one of these brackets, I can't do anything about the $\displaystyle P_n(...)$. I'm stuck :/