Roots of polynomial equation $2a x^\gamma + ax^{\gamma - 1} - 2 = 0$

Question

I would like to find roots of the following polynomial equation $$2a x^\gamma + ax^{\gamma - 1} - 2 = 0$$ where $a,\gamma>0$ (we might also assume that $\gamma \in\mathbb{N}$ if needed). Playing a little in Wolfram, I conjecture that for $\gamma \in \mathbb{N}$, there seems to a one positive and one negative real root, is it possible to find explicit solutions in terms of $a$?

Answer 1

First, note that $\gamma$ must be an integer for the l.h.s. to be a polynomial.

And no, it is not in general possible to solve explicitly for the roots, at least not in terms of radicals. For example: Taking $a = 1$, $\gamma = 5$ gives the polynomial $$2x^5 + x^4 - 2;$$ its Galois group is isomorphic to $S_5$, which is not solvable, and hence its roots cannot be written in terms of radicals. Thus, there is no such formula for the roots in terms of $a$.