## Solutions to functions using Liouville's Theorem

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

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

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