Suppose you have a rational function, of the form H ( x ) = B

Jayla Faulkner

Jayla Faulkner

Answered question

2022-05-17

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".

Answer & Explanation

Timothy Mcclure

Timothy Mcclure

Beginner2022-05-18Added 15 answers

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.

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