Suppose you have a rational function, of the form H(x)=B(x)/A(x). It's differential is of the form H'(x)=(B'(x)*A(x)-B(x)*A'(x))/(A(x))^2. It is trivial to prove that, if k is a root of B(x) or A(x) with rank r, r>=2, then k is also a root of H'(x)'s numerator, B'(x)*A(x)-B(x)*A'(x), with rank at least r-1. Is there a theorem that states this exact thing?
The reason I need this is, I am going to take an exam soon, and in a specific part of the exam this observation would be very useful. But I will have to prove it by hand during the exam, and time will be very limited, so I was wondering if there is already a name for this to reference it directly and get on with it.
EDIT: Had erroneously named the desired form "polynomial fraction", instead of the right "rational function".
The name for "polynomial fractions" is rational functions.
I don't think there's a special name for this result. Closely related is the fact that a polynomial and its derivative are relatively prime if and only if there's no repeated root.
Anyway, I would just state it and use it when necessary.