Using partial Fraction

Question

I want to apply partial fraction process to the fraction below :

$$(*) \frac{s^2-4s+1}{(s-1)^4}$$

let $$ \frac{A}{s-1}+\frac{B}{(s-1)^4} = (*)  \ then \\ A(s-1)^3+B = s^2-4s+1 \\ let \ s=1 \\ \implies B=1-4+1 = -2 \\ now \ let \ s=2 \\ A+B = 4-8+1 = -3 \ $$

So $A=-1$ and $B=-2$ But

$$-1(s-1)^3 -2 \neq s^2-4s+1$$

Why? Am I thinking wrong? What mistake I'm doing here.

Answer 1

The denominator has degree $4$, so the numerator has degree $3$. This means your setup should be either

$$\frac{AS^3+BS^2+Cs+D}{(s-1)^4}$$

or

$$\frac{A}{s-1} + \frac{B}{(s-1)^2}+ \frac{C}{(s-1)^3}+ \frac{D}{(s-1)^4}$$

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Either of the two options works, the second one is more useful though.