I went through the proof of the Laurent phenomenon for Cluster Algebras in Fomin and Zelevinsky's initial paper: Cluster Algebras I: Foundations. I am stuck at their claim that the gcd of two exchange polynomials is 1.

I am working in a setup, where all coefficients are set to 1, thus the coefficient group is just the group of integers (therefore the cluster algebras are of geometric type). But why do the exchange polynomials have to be different? In my opinion their gcd could be a binomial in the cluster variables. This would not be a unit in the ring of Laurent polynomials. What am I missing?

Cheers