The question is based on
general polynomial of degree 5 or higher have "no closed-form formula"
This is not exactly true, what it should be said is that "general algebraic equations of degree higher than 4 do not admit solutions by radicals" which means that they cannot be solved by operations implying combinations of ordinary additions, multiplications, divisions, raising to powers, root extractions...
On the other side it was shown by Hermite that fifth degree equations can be solved using the modular elliptic functions, which provide a generalization of the so called trigonometric solution of eqs. with degree lower than 5.
Higher order (than 5) algebraic equations can be solved by employing other forms of elliptic functions.
In more recent times the use of the Lagrange inversion formula has allowed solutions in terms of hypergeometric functions.
This technique was developed by an italian mathematician G. Belardinelli in 1959 and later rediscovered by M. L. Glasser in 2000 J. Comp. and Appl. Math. 118 (2000) 169-171.