Let's consider complex points $a_1,a_2,..., a_n \in \mathbb C$, and polynomial $p$ of degree $\deg p \ge n \ge 1$. I want to prove that function $q$:
$$q(z) = p[z, a_1,a_2,...,a_n]$$
has to be polynomial of degree $\deg p - n$.
My solution
From mean value theorem, we know that:
$$\forall_{z \in \mathbb C} \exists_{\epsilon_z \in (\min(z, a_1,...,a_n), \max(z,a_1,...,a_n))}: p[z, a_1,...,a_n] = \frac{p^{(n)}(\epsilon_z)}{n!}$$
Now, since $p$ is polynomial of degree $\deg p$, then $p^{(n)}$ will be polynomial of degree $\deg p - n$, and this is sufficient fact to say that $q$ has to be polynomial of degree $\deg p - n$.
Is my justification correct?