I wish to show
$A=\dfrac{\mathbb{C}[x,y,z]}{x^5+y^3+z^2}$ is UFD
What I have tried so far:
I was thinking of applying Nagata's criterion. Indeed, $-(x^5+y^3)$ cannot be a square in $\mathbb{C}[x,y]$. Hence $A$ is a domain. Note that $$\frac{A}{xA}=\frac{\mathbb{C}[x,y,z]}{(x,x^5+y^3+z^2)} \cong \frac{\mathbb{C}[y,z]}{y^3+z^2}$$ which is a domain as $-y^3$ is not a square in $\mathbb{C}[y]$. Hence $x$ is a prime in $A$. Thus enough to check $A[\frac1x]$ is a UFD. Now I am stuck here. I cannot simplify the ring $A[\frac1x]$.
Any help/ suggestions to tackle this problem.