I single variable polynomial splits completely in some field extension. Say $f(x,y,z)$ has a root $(a_0, b_0, c_0) \in \Bbb{Q}^3$.
In a single variable we can say that if $a_0$ is a root then the linear polynomial $x-a_0$ divides $f$. In particular the polynomial $x-a_0 = 0 \iff x = a_0$. In several variables though there are polynomials that satisfy that, namely:
$p(x,y,z) = (x- a_0)^{2k} + (y-b_0)^{2k} + (z-c_0)^{2k}$ for some integer $k \geq 1$.
I.e. $p(x,y,z) = 0 \iff (x,y,z) = (a_0, b_0, c_0)$. So if the latter is a root of $f$, does the polynomial $p$ have any useful relation to $f$?
I'm aware of how to compute Grobner bases. I don't know how this relates though.
I think the proper generalization is as follows: if $p=p(X_1,X_2,\ldots,X_n)$ is a polynomial over an algebraically closed field $F$ and $q$ is a nonzero polynomial over $F$ in $n$ variables which is irreducible, and such that $q(X)=0\implies p(X)=0$, then $q$ divides $p$. This follows from Hilbert's Nullstellensatz.
To see that this is a generalization, notice that for polynomials of single variable over algebraically closed fields, the irreducible polynomials are exactly the polynomials of degree $1$ and a polynomial in one variable of degree $1$ has only a single point in its zero set.