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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.