closed-form expression for roots of a polynomial

Solution 1:

As you might already know, solutions to the quintic can be expressed in terms of either ${}_4 F_3$ hypergeometric functions or Jacobi theta functions. See King or Prasolov/Solovyev for details.

For polynomials of higher degree, there is also a general formula for the roots, due to Umemura. The formulae involve the multidimensional generalization of the Jacobi theta functions (the Riemann theta function), and are a bit unwieldy; see Umemura's paper if you want more details. See also this preprint for a solution of the reduced polynomial equation $x^n-x-\alpha=0$ in terms of hypergeometric functions.

Solution 2:

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. Appl. Math. 118 (2000) 169-171.

Solution 3:

Polynomial equations can be solved with the 4 arithmetic operations, prime power roots and a suitable collection of other special roots obtained from a subset of the "unsolvable" equations of 5 and higher order. For the quintic equation, if I recall correctly, one needs only add the "star-root" x = *√c to x⁵ + x = c of real numbers c to the list of operations.

That's described in passing in R. Bruce King, Beyond the Quartic Equation, Birkhäuser, Boston, Basel, Berlin, 1996.