Assume that $f_n$ is monic. Then for indeterminates $u_1,\ldots,u_{n-1}$, all coefficients of the polynomial $Res_x(f_n,u_1f_1,\ldots, u_{n-1}f_{n-1})\in k[u_1,\ldots,u_{n-1}]$ vanish if and only if $f_1,\ldots,f_n$ have a common root [expanding out this resultant then gives the list of polynomials].

This is how elimination theory works - if $f_1(x_1,\ldots,x_m)=0,\ldots, f_n(x_1,\ldots,x_m)=0$ set-theoretically define a variety $V$, $f_n$ is monic in $x_m$ (which one ensure by applying a Noether normalization coordinate change), and $\pi:\mathbb{A}^m\to \mathbb{A}^{m-1}$ is the projection away from the last coordinate, then $\pi(V)$ is a variety in $\mathbb{A}^{m-1}$ defined by the coefficients of $Res_x(f_n,u_1f_1,\ldots, u_{n-1}f_{n-1})$. Repeatedly eliminating variables like this eventually gives you a finite morphism surjecting $V$ to $\mathbb{A}^{\dim(V)}$.