What's the logic behind partial fraction decomposition?

[A.MHF]

Ok so I took partial fraction decomposition in Calc II, and now I'm taking it again in Differential Equations course. The problem is that I don't really understand what I'm doing.
I understand the procedure when having simple real roots, for example

2x+1/(x+1)(x+2), it becomes A/(x+1) + B/(x+2)
Because multiplying the two would get us a common denominator of (x+1)(x+2), which is what we want.

But I don't understand why when having repeated roots we have to include all the powers in the expansion?
For example:

2x+1/(x+1)^3= A/(x+1)+ B/(x+1)^2 + C/(x+1)^3

Also, when having irreducible factor we have to put (Ax+B) in the numerator instead of just "A".
Can someone help me understand what's going on here?

Thanks.

 

[Orodruin]

It is a matter of having enough constants available to adapt the polynomial you obtain when putting everything with a common denominator to the original polynomial.

 

[HallsofIvy]

The logic in writing (2x+ 1)/(x+ 1)(x+ 2) as A/(x+ 1)+ B/(x+ 2) (Note the parentheses. This is NOT "2x+ 1/(x+1)(x+2)" which cannot be written in that form) is the other way:

A/(x+ 1)+ B/(x+ 2)= [A(x+ 2)]/(x+1)x+2)+ [B(x+ 1)]/(x+ 1)(x+ 2)= (Ax+ 2A+ Bx+ B)/(x+ 1)(x+ 2)= [(A+ B)x+ (2A+ N)]/(x+1)(x+2) which will be equal to (2+ 1)/(x+1)(x+2) as long as A+ B= 2 and 2A+ B= 1. And, of course, A= -1, B= 3 satisfy tose.