Let $k$ be an infinite field, not algebraically closed. Let $\operatorname{Specm} k[x_1,\cdots,x_n]$ be the set of maximal ideals of $k[x_1,\cdots,x_n]$ and define a topology on $\operatorname{Specm} k[x_1,\cdots,x_n]$ in which the closed sets are of the form $Z(E) = \left\{m \in \operatorname{Specm} k[x_1,\cdots,x_n]: E \subseteq m \right\}$, where $E$ is a subset of $k[x_1,\cdots,x_n]$. Now, let $X$ be the subset of $\operatorname{Specm} k[x_1,\cdots,x_n]$, consisting of maximal ideals of the form $(x_1-c_1,\cdots,x_n-c_n)$ with $c_i \in k$. I need a hint towards showing that $X$ is dense in $\operatorname{Specm} k[x_1,\cdots,x_n]$. I proved this for the case where $k=\mathbb{R}$ and $n=1$, but i made decisive use of the fact that $\mathbb{R}[x]$ is a principal ideal domain. The case $\mathbb{R}[x_1,x_2]$ seems to be fundamentally different.
PS: Please, only a hint, not the solution. Thanks.