## Task B-4.3.

Determine all polynomials $P$ with real coefficients for which the equality

$$
x \cdot P(x-1)=(x-2023) \cdot P(x)
$$

holds for all real numbers $x$.
## Solution.

Since the equality $x \cdot P(x-1)=(x-2023) \cdot P(x)$ holds for all real numbers $x$, by substituting $x=0$ we get $0 \cdot P(-1)=-2023 \cdot P(0)$, from which it follows that $P(0)=0$. Furthermore, for $x=1$ we get $1 \cdot P(0)=-2022 \cdot P(1)$, from which it follows that $P(1)=0$. By substituting $x=2$ into the given equality, we get $2 \cdot P(1)=-2021 \cdot P(2)$, so $P(2)=0$.

Inductively, for $x=2022$ we get that $2022 \cdot P(2021)=-1 \cdot P(2022)$, so $P(2022)=0$.

Since $0,1,2, \ldots, 2022$ are the roots of the polynomial $P$, the polynomial $P$ can be written in the form $P(x)=$ $x \cdot(x-1)(x-2) \cdots(x-2022) \cdot Q(x)$, for some polynomial $Q$.

If we now substitute this expression for the polynomial $P$ into the initial equality, we get

$x \cdot(x-1)(x-2) \cdots(x-2022) \cdot(x-2023) \cdot Q(x-1)=(x-2023) \cdot x \cdot(x-1)(x-2) \cdots(x-2022) \cdot Q(x)$, from which it follows that $Q(x-1)=Q(x)$ for all real numbers $x$. The only polynomials with this property are constants. Therefore, $Q(x)=c$, for some $c \in \mathbb{R}$.

All polynomials with the desired property are of the form

$$
P(x)=c \cdot x \cdot(x-1)(x-2) \cdots(x-2022), c \in \mathbb{R}
$$