For what non-zero integers $a,b,c$ (not just positive, but negative, too!) does:
$$f\left(x\right)=72x^{a}-12x^{b}-2x^{c}-27$$
have a rational zero $x_{0}\in\mathbb{Q}$?
What is this zero, as a function of $a,b,c$—if such a function can be found?
(It would be ideal to have non-zero integer values of $a,b,c$ for which $x_{0}$ can be expressed in closed form in terms of $a,b,c$.)
Addendum: $a=b=c=1$ and $a=b=c=-1$ both work, but are there any others?