Question 121
Let $P(X)$ be a monic polynomial with real coefficients of degree $2022$. Emile plays the following game: he writes the polynomial $P(X)$ on the board and at each step, if the polynomial $f(X)$ is written on the board, Emile may replace it by:
- the polynomial $f(X)+c$, for a real number $c$ of his choice, or
- the polynomial $P(f(X))$.
Find all non-negative integers $n$ such that, whatever the initial polynomial $P$, Emile can find a sequence of operations such that the resulting polynomial has exactly $n$ distinct real roots.
- the polynomial $f(X)+c$, for a real number $c$ of his choice, or
- the polynomial $P(f(X))$.
Find all non-negative integers $n$ such that, whatever the initial polynomial $P$, Emile can find a sequence of operations such that the resulting polynomial has exactly $n$ distinct real roots.