Finding Residuals A, B & C: Step-by-Step Guide

[Ry122]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

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.

 

[Ry122]

thanks

 

[Ry122]

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