I have 2 questions:

1. What is the reasoning for having to have separate fractions like: A/(x-b) + B/(x-b)^2 + C/(x-b)^3 + D/x to do a partial fraction decomposition of p(x)/x[(x-b)^3] ?

2. [solved] Lets say you have a polynomial P(x)=ax^3 + bx^2 + cx +d and you divide it by x-b and you get a remainder (r). You can now write the polynomial as Q(x)=(x-b)(ex^2 + fx + g + (r/x-b)). Since Q(x)=P(x) they should have the same zeros right? But this isn't always true because that would mean b must be a zero but this is not necessarily true.

Basically, I am saying that according to a theorem--I think it is called the factor theorem--if x-b is factor of P(x) then b is a zero of P(x). But the problem is that you can divide a second degree or greater polynomial P(x) by any polynomial x-b even if b is not a factor of the polynomial, and just write the remainder (r) as (r/x-b) inside the quotient, and those polynomials should be equal but their zeros are not necessarily equal.