To a mathematician: Where can I read about deflation (gcd(S,dS/dx)) for a first time learner?

I just bought a new book, but it includes things that I have never covered. It mentions the deflation of a polynomial S as gcd(S,dS/dx) and squarefree factors. It also mentions fractions of the form Q/V^k where V is squarefree. It says you should be able to find B and C such that Q/(1-k) = B(dV/dx) + CV and deg(B) is less than deg(V). I don't know why you should assume there is such a B and C. I think I'm missing some mathematical background, and I would like to know what category of math and what type of books would be good to read for a first time learner about this stuff, mainly the deflation stuff, so I can understand how they can up with these things. I've had multivariable calculus and a mathematical physics book to play with but I'm not strong at number theory, abstract algebra, or linear algebra maybe. I just want to know what category to study and what type of books to buy.