## Solutions to functions using Liouville's Theorem

Question
If $a\in R$ is the root of an irreducible polynomial $d\geq 2$, with coefficients in $Q$, then there is a $\delta>0$ such that
$|a-\frac{p}{q}|>\frac{\delta}{q^{2{\sqrt{d}}}}$
for all $\frac{p}{q} \in Q$

Use this to prove that if
$f(X,Y)=a_dX^d+a_{d-1}X^{d-1}Y+a_{d-2}X^{d-2}Y^2+...+a_1XY^{d-1}+a_0Y^d$

where $a_i$ $\in{Z}$, $d\geq5$ and $f(x,1)$ $\in{Z[x]}$ is irreducible
then $F(X,Y)=1$ has only finitely many solutions $X,Y$ $\in{Z}$

Request
I need a method of approaching this question. I am supposed to use the facts that the exponents of X and Y always sum to d and $f(p,q)=q^d(f(\frac{p}{q},1))$