Show that a polynomial has at least 1 real root

Question

I have the polynomial $P(x)=x^{2}+2013x+1$ and a number $n\in\mathbb{N}$. Now I have to show that the polynomial $P(P(...P(x)...)$ $(n$ times$)$ has at least one real root. How can I do this?

Answer 1

Note that the quadratic equation $$P(x)=x$$ has some real root $x_-<0$.

For every $n\ge 1$, denote the $n$-th iteration of $P$ by $P^{\circ n}$. Then $$P^{\circ n}(x_-)=x_-<0\quad\text{and}\quad P^{\circ n}(+\infty)=+\infty,$$ so there exists $x_n\in(x_-,+\infty)$, such that $P^{\circ n}(x_n)=0$.

Answer 2

The polynomial has two real zeros, call them $a$ and $b$, where $a<b$ (this we can tell from taking the discriminant). If you can show that $P([a,b])\supset [a, b]$, then you're home free, because then $a$ has a preimage $c_1\in [a,b]$, then $c_1$ has a preimage $c_2\in [a,b]$, then...