I am trying to understand the proof of Laurent phenomenon of cluster algebras in [the book (Sergey Fomin, Lauren Williams, Andrei Zelevinsky, *Introduction to Cluster Algebras. Chapters 1-3*, arXiv:1608.05735v1)](https://arxiv.org/abs/1608.05735). 


On page 45, it is said that "We see that $x\_j'$ is linear in $x\_q$ and hence irreducible (as a Laurent polynomial in $\tilde{\bf x}$), i.e., it cannot be written as a product of two non-monomial factors." Here 
\begin{align}
x\_j' = x\_j^{-1} (x\_k^c x\_q M\_1 + M\_2),
\end{align}
where $M\_1, M\_2$ are monomials in $x\_i$'s ($i \not\in \{j,k,q,r\}$). 


Why $x\_j'$ cannot be written as a product of two non-monomial factors. Maybe we can have something like
\begin{align}
x\_j' = (x\_q+x\_r)(x\_j+1)?
\end{align}
Here $(x\_q+x\_r)(x\_j+1)$ is linear in $x\_q$ and $(x\_q+x\_r)(x\_j+1)$ is a product of two non-monomial factors.


Where do I made a mistake? Thank you very much.



Trying to understand the proof of Laurent phenomenon of cluster algebras
You need to use the fact that $x'\_j = M x\_q + M'$, where $M$ and $M'$ are Laurent *monomials* in the remaining variables. Their observation is that if $x'\_j$ factored, it would be as $x'\_j = (P\_1 x\_q + P\_2)P\_3$, where $P\_1, P\_2, P\_3$ are Laurent *polynomials* in the remaining variables. But then $P\_1 P\_3 = M$ and $P\_2 P\_3 = M'$, which forces each of $P\_1, P\_2, P\_3$ to be Laurent *monomials* in the remaining variables. So there is no non-trivial factorization of $x'\_j$.


