Possible for variable to cancel in a product of multivariate rational expressions?

Question

Let $f,g,p_i,q_i$ be polynomials over some field with $\gcd(p_i,q_i)=1$ and $q_i$ are not constants for $i=1,2$. Assume that one or more of $p_i$ or $q_i$ has a term containing a variable $x$ not present in $f$ or $g$, does the simplified product

$$(f+\frac{p_1}{q_1})(g+\frac{p_2}{q_2})$$

necessarily contain the variable $x$?

Answer 1

\[\left(1 + \frac{1}{x}\right)\left(1-\frac{1}{1+x}\right) = 1\]