Well, a "naïve" answer could be based on the fact that a general method for finding the solutions of a system of simultaneous polynomial equations $$ f_1(x)=0, \ \ f_2(x)=0, \ \ \cdots \ \ f_n(x)=0 $$ is finding the $\gcd\bigl(f_1, f_2, \cdots, f_n\bigr)=g(x)$, of these polynomials.

Assuming that we are speaking about polynomials with complex coefficients and since $\mathbb{C}[x]$ is a PID, the ideal of $\mathbb{C}[x]$ generated by the polynomials $f_1, f_2, \dotsc, f_n$ is the ideal generated by their greatest common divisor: $$ \langle f_1, f_2, \dotsc, f_n\rangle=\langle\gcd\bigl(f_1, f_2, \dotsc, f_n\bigr)\rangle=\langle g\rangle. $$

Thus, the solution of the initial system, is equivalent to the solution of the single equation $$ g(x)=0. $$

And well, ok maybe there is no explicit formula for the gcd of a number of univariate polynomials but there is certainly an algorithm: The $\gcd\bigl(f_1, f_2, \dotsc, f_n\bigr)$, can be found using the Euclidean division algorithm for finding the $\gcd$ of two univariate polynomials in $\mathbb{C}[x]$ together with the fact that for $n\geq 3$ $$ \gcd(f_1, f_2, \dotsc, f_n)=\gcd\bigl(f_1,\gcd(f_2, \dotsc, f_n)\bigr). $$

P.S.: I am not sure if this satisfies you as an answer but I was led to this description driven by the way the question has been posed and the subsequent discussion in the comments.

Well, the "naïve" answer (already implied in @Somnium's comment) is that a general method for finding the solutions of a system of simultaneous polynomial equations $$ f_1(x)=0, \ \ f_2(x)=0, \ \ \cdots \ \ f_n(x)=0 $$ is finding the $\gcd\bigl(f_1, f_2, \cdots, f_n\bigr)=g(x)$, of these polynomials:

Assuming that we are speaking about polynomials with complex coefficients and since $\mathbb{C}[x]$ is a PID, the ideal of $\mathbb{C}[x]$ generated by the polynomials $f_1, f_2, \dotsc, f_n$ is the ideal generated by their greatest common divisor: $$ \langle f_1, f_2, \dotsc, f_n\rangle=\langle\gcd\bigl(f_1, f_2, \dotsc, f_n\bigr)\rangle=\langle g\rangle. $$

Thus, the solution of the initial system, is equivalent to the solution of the single equation $$ g(x)=0. $$

Maybe there is no explicit formula for the gcd of a number of univariate polynomials (in terms of their coefficients) but there is certainly an algorithm: The $\gcd\bigl(f_1, f_2, \dotsc, f_n\bigr)$, can be found using the Euclidean division algorithm for finding the $\gcd$ of two univariate polynomials in $\mathbb{C}[x]$ together with the fact that for $n\geq 3$ $$ \gcd(f_1, f_2, \dotsc, f_n)=\gcd\bigl(f_1,\gcd(f_2, \dotsc, f_n)\bigr). $$

P.S.: I am not sure if this satisfies you as an answer but I was led to this description driven by the way the question has been posed and the subsequent discussion in the comments.

Well, a "naive" answer could be based on the fact that a general method for finding the solutions of a system of simultaneous polynomial equations $$ f_1(x)=0, \ \ f_2(x)=0, \ \ ... \ \ f_n(x)=0 $$ is finding the $\gcd\big(f_1, f_2, ..., f_n\big)=g(x)$, of these polynomials.

Assuming that we are speaking about polynomials with complex coefficients and since $\mathbb{C}[x]$ is a PID, the ideal of $\mathbb{C}[x]$ generated by the polynomials $f_1, f_2, ..., f_n$ is the ideal generated by their greatest common divisor: $$ <f_1, f_2, ..., f_n>=<\gcd\big(f_1, f_2, ..., f_n\big)>=<g> $$

Thus, the solution of the initial system, is equivalent to the solution of the single equation $$ g(x)=0 $$

And well, ok maybe there is no explicit formula for the gcd of a number of univariate polynomials but there is certainly an algorithm: The $\gcd\big(f_1, f_2, ..., f_n\big)$, can be found using the Euclidean division algorithm for finding the $\gcd$ of two univariate polynomials in $\mathbb{C}[x]$ together with the fact that for $n\geq 3$ $$ \gcd(f_1, f_2, ..., f_n)=\gcd\big(f_1,\gcd(f_2, ..., f_n)\big) $$

P.S.: I am not sure if this satisfies you as an answer but i was led to this description driven by the way the question has been posed and the subsequent discussion in the comments.

Well, a "naïve" answer could be based on the fact that a general method for finding the solutions of a system of simultaneous polynomial equations $$ f_1(x)=0, \ \ f_2(x)=0, \ \ \cdots \ \ f_n(x)=0 $$ is finding the $\gcd\bigl(f_1, f_2, \cdots, f_n\bigr)=g(x)$, of these polynomials.

Assuming that we are speaking about polynomials with complex coefficients and since $\mathbb{C}[x]$ is a PID, the ideal of $\mathbb{C}[x]$ generated by the polynomials $f_1, f_2, \dotsc, f_n$ is the ideal generated by their greatest common divisor: $$ \langle f_1, f_2, \dotsc, f_n\rangle=\langle\gcd\bigl(f_1, f_2, \dotsc, f_n\bigr)\rangle=\langle g\rangle. $$

Thus, the solution of the initial system, is equivalent to the solution of the single equation $$ g(x)=0. $$

And well, ok maybe there is no explicit formula for the gcd of a number of univariate polynomials but there is certainly an algorithm: The $\gcd\bigl(f_1, f_2, \dotsc, f_n\bigr)$, can be found using the Euclidean division algorithm for finding the $\gcd$ of two univariate polynomials in $\mathbb{C}[x]$ together with the fact that for $n\geq 3$ $$ \gcd(f_1, f_2, \dotsc, f_n)=\gcd\bigl(f_1,\gcd(f_2, \dotsc, f_n)\bigr). $$

P.S.: I am not sure if this satisfies you as an answer but I was led to this description driven by the way the question has been posed and the subsequent discussion in the comments.