[Math] 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 $P^{\circ n}(x_n)=0$.