Consider the case in which one of the roots is repeated:
In this fraction the denominator polynomial has a repeated root at s=-a. The remainder of the denominator polynomial is called D'(s); it has no roots as s=-a. The numerator polynomial is N(s). If we expand this fraction we get
The term N'(s)/D'(s) represents the expansion of all of the terms except those with roots at s=-a.
We can find A1 by multiplying by (s+a)2 and setting s=-a (i.e., the cover-up method).
To find A2 we note that the we can get rid of the A1 term by differentiating the result above
Now we can solve for A2 by setting s=-a.
Now consider the case of multiply repeated roots (n>2).
The resulting partial fraction expansion is
As before we can easily find A1 and A2
If we continue to differentiate we can find the other coefficients in a similar manner