smc

$P(x)=x^{2022}+x^{1011}+1$ is equal to $\frac{x^{3033}-1}{x^{1011}-1}$ by difference of powers. Therefore, the answer is a polynomial that divides $x^{3033}-1$ but not $x^{1011}-1$. Note that any polynomial $x^m-1$ divides $x^n-1$ if and only if $m$ is a factor of $n$. The prime factorizations of $1011$ and $3033$ are $3\ast 337$ and $3^2\ast 337$, respectively. Hence, $x^9-1$ is a divisor of $x^{3033}-1$ but not $x^{1011}-1$. By difference of powers, $x^9-1=(x^3-1)(x^6+x^3+1)$. Therefore, the answer is $\boxed{x^6+x^3+1}$.