Solving $x^p+ax^q+b=0$ with $x,a,b$ integer and $p-q>1$

Question

Help solving $x^p+ax^q+b=0$, where $p,q,x \geq 0$ and $a,b \in\mathbb{Z}$.

I am well aware of the complexity of this equation. However, I am mostly interested in the following particular case: Given that $x$ is a non-negative integer, if $p-q>1$, can't I derive logically that $x=b=0$?