# Why are these two polynomials not coprime?

Let $P(s)$, $Q(s)$, $A(s)$, and $B(s)$ be polynomials of a complex variable such that $P(s)/Q(s)=A(s)/B(s)$.

The polynomials have degree as follows: degree $P(s)\le n$, degree $Q(s)=n$, degree $A(s) \le r$, degree $B(s) = r$, and $n>r$.

Why does this imply that the polynomials are not coprime?

This last fact is stated in a textbook with no explanation as to why. For context, this is a proof in control theory regarding controllability of LTI system in the Laplace domain.

• Intuitively: we must have been able to cancel a common factor from both $P$ and $Q$ – Omnomnomnom Aug 23 '16 at 5:05

A rational function, of total (numerator minus denominator) degree $\leq 0$, has been expressed in two different, but equal, ways as a polynomial fraction, with denominators of different degrees.
At most one of these two ways is reduced, and it would have to be the one of lower denominator degree. The other way, $P/Q$, must have some shared factors of the numerator and denominator.
This argument did not need the information that total degree is at most $0$.