# What's the logic behind partial fraction decomposition?

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
Staff Emeritus
Homework Helper
Gold Member
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