# Finding residual

1. Jun 14, 2012

### Ry122

1. The problem statement, all variables and given/known data

Given the following working, what steps must be taken to determine the values of the residuals A, B, and C?

2. Relevant equations

3. The attempt at a solution

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2. Jun 14, 2012

### HallsofIvy

Staff Emeritus
You have
$$\frac{7}{s(s- 2)^2}= \frac{A}{s}+ \frac{B}{s- 2}+ \frac{C}{(s- 2)^2}$$

Multiply on both sides by $s(s- 2)^2$ to get
$$7= A(s- 2)^2+ Bs(s- 2)+ Cs$$

Now you can
1) multiply the right side and combine "like powers of s":
$$7= As^2+ 4As+ 4A+ Bs^2- 2Bs+ Cs= (A+ B)s^2+ (4A- 2B+ C)s+ 4A$$
for all s so we must have A+ B= 0, 4A- 2B= 0, and 4= 7, three equations to solve for A, B, and C.
or
2) Choose any three values for s you want so as to get three equations to solve for A, B, and C. For example, since we have the term s-2, taking s= 2 reduces to just 2C= 7. Since we have the term s, taking s= 0 reduces to 4A= 7. Taking s= 1, just because it is an easy number, A- B+ C= 7.

3. Jun 14, 2012

### Ry122

thanks

4. Jun 14, 2012

### Ry122

why was the right most term duplicated onto the LHS?
EDIT: nevermind, i worked it out