Can a generalized root formula exist for polynomials with finite degrees? [closed]

Question

Let $p\in\mathbb{Z}[x]$, set $d = \textbf{Deg}(p)$, and write $$p(x) = \sum_{i=0}^{d}{a_ix^i}$$ for some sequence $\{a_0,a_1,a_2,...., a_{d-1}\}$. Is there a mapping $\mathcal{F}$ so that $$\mathcal{F}\biggl(\{a_0,a_1,a_2,...., a_{d-1}\}\biggr) = \{x:~ \sum_{i=0}^{d}{a_ix^i} = 0 \}.$$ That is does a general formula for the roots of every polynomial exist. The Quadratic formula is known and we can write the roots of $a_3x^2+a_2x+a_1$ as $$\dfrac{-a_2\pm \sqrt{a_2^2-4a_3a_1}}{2a_3}.$$ So we know when $d=3$ the function $\mathcal{F}$ at $\{a_1,a_2,a_3\}$ is $\dfrac{-a_2\pm \sqrt{a_2^2-4a_3a_1}}{2a_3}$. There might exist a function in the form of a quotient $$\mathcal{F}(\{a_0,a_1,a_2,...,a_{d-1}\})= \dfrac{f(a_0,a_1,a_2,...,a_{d-1})}{g(a_0,a_1,a_2,... ,a_{d-1})}.$$ When $d=2$ we have $f:= -a_2\pm \sqrt{a_2^2-4a_3a_1}$ and $g:= 2a_3$ is a solution.