This is most likely simple, but I want to ask a small question.
Let $k$ be a field, $k(x)$ the field of rational functions in $x$. Let $t\in k(x)$ be the rational function $P(x)/Q(x)$ with relatively prime polynomials $P(x),Q(x)\in k[x]$, with $Q(x)\neq 0$.
Let $P(X)-tQ(X)$ be a polynomial in the variable $X$ and coefficients in $k(t)$.
I want to show that the degree of $P(X)-tQ(X)$ as a polynomial in $x$ with coefficients in $k(t)$ is the maximum of the degrees of $P(X)$ and $Q(X)$.
If $P(X)$ and $Q(X)$ have different degrees, this is clear. Otherwise, I suppose both have degree $n$, and the leading term of $P(X)$ has form $k_1X^n$, and the leading term of $Q(X)$ has form $tk_2X^n$. I only want to show that these terms do not cancel. If they did, then $k_1/k_2=t$, so $t$ is a constant rational function. Is there a contradiction to be had, so that this contradictory case cannot actually happen?