I know that in general polynomials above degree 4 do not have analytic solutions, except for a few special cases. What I want to know is whether this particular polynomial is one of those cases. The equation is
$$ax^p+bx^q+c=0$$
where p and q are both positive integers. One of them equals 4, if that helps at all.
Yes, in terms of hypergeometric functions in one variable. To quote,
"... This technique gives closed-form solutions in terms of hypergeometric functions in one variable for any polynomial equation which can be written in the form $x^p+bx^q+c = 0$..."
See (eq.42) of this Mathworld entry.
P.S. If in radicals, and $x^p+bx^q+c = 0$ is irreducible (and I assume $p,q$ are co-prime), "...Guralnick and Shareshian proved that if the degree of the equation is $d\ge7$ (except $d=8$), then there are only finitely many equivalence classes of irreducible degree $d$ trinomials in $\bf{Q}[x]$ whose Galois group is solvable...". See the answer to my MathOverflow question about septic trinomials $x^7+ax+b=0$.
