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))