Factorizing a cubic polynomial in 3 variables.

Question

I recently saw a question in which we had to factorize a cubic polynomial in 3 variable. The method shown in the answer asked me to assume any 2 variables as constants and 1 as a variable and then follow the factor theorem. Just then a question struck my mind about the expression :-- $$(xyz-1)$$ I tried to factorise it by the above method. First I assumed $x$ as variable, then $y$ and then $z$ as variable and ended up with :-- $$(x-\frac{1}{yz})(y-\frac{1}{xz})(z-\frac{1}{yx})$$ And I had factorised it in 3 factors because the polynomial was cubic. The above 3 factors satisfy the expression through factor theorem but when we actually multiply the 3 factors we end up having some thing else. Please tell me where I went wrong. It would be a great help. $$Thank you$$