4A. Find all polynomials

$$
P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}, n \geq 2
$$

with real and non-zero coefficients such that $P(x)-P_{1}(x) P_{2}(x) \cdot \ldots \cdot P_{n-1}(x)$ is a constant polynomial, where

$$
P_{1}(x)=a_{1} x+a_{0}, P_{2}(x)=a_{2} x^{2}+a_{1} x+a_{0}, \ldots, P_{n-1}(x)=a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}
$$
Solution. Since the polynomial $P(x)-P_{1}(x) P_{2}(x) \cdot \ldots \cdot P_{n-1}(x)$ is constant and $a_{k} \neq 0, k=\overline{1, n-1}$, we have that

$$
\begin{aligned}
\operatorname{deg} P=\operatorname{deg}\left(P_{1} P_{2} \ldots \cdot P_{n-1}\right) & =\operatorname{deg} P_{1}+\operatorname{deg} P_{2}+\ldots+\operatorname{deg} P_{n-1} \\
& =1+2+\ldots+(n-1)=\frac{n(n-1)}{2}
\end{aligned}
$$

We get that $n=\frac{n(n-1)}{2}$, so $n \in\{0,3\}$. Since $n \geq 2$, we conclude $n=3$. Therefore, $P(x)=a_{3} x^{3}+a_{2} x^{2}+a_{1} x+a_{0}$, and $P(x)-P_{1}(x) P_{2}(x)=k, k \in \mathbb{R}$ if and only if:

$$
\begin{aligned}
& a_{3}=a_{1} a_{2} \\
& a_{2}=a_{1}^{2}+a_{0} a_{2} \\
& a_{1}=2 a_{0} a_{1} \\
& a_{0}=a_{0}^{2}+k
\end{aligned}
$$

From (1) we have $a_{1}\left(1-2 a_{0}\right)=0$, since $a_{1} \neq 0$ it follows that $a_{0}=\frac{1}{2}$ and $k=\frac{1}{4}$. Let $a_{1}=a \in \mathbb{R} \backslash\{0\}$. Then $a_{2}=2 a^{2}$ and $a_{3}=2 a^{3}$.

Thus, the required polynomials are of the form

$$
P(x)=2 a^{3} x^{3}+2 a^{2} x^{2}+a x+\frac{1}{2}, \text { where } a \in \mathbb{R} \backslash\{0\}
$$