Express F (s) in partial fractions:
2
6 s + 2 s - 38
F(s) = ---------------
(s-1)(s+1)(s+2)
2
6 s + 2 s - 38 a b c
--------------- = --- + --- + ---
(s-1)(s+1)(s+2) s-1 s+1 s+2
All three denominators are linear factors that are not repeated.
The cover-up rule may be used in all three cases.
2
6(1) + 2(1) - 38 -30
a = ----------------- = --- = -5
*** (1+1)(1+2) 6
2
6(-1) + 2(-1) - 38 -34
b = ------------------- = --- = +17
(-1-1) *** (-1+2) -2
2
6(-2) + 2(-2) - 38 -18
c = ------------------- = --- = -6
(-2-1)(-2+1) *** 3
Therefore
2
6 s + 2 s - 38 -5 17 6
F(s) = --------------- = --- + --- - ---
(s-1)(s+1)(s+2) s-1 s+1 s+2
If one does not wish to employ the cover-up rule, then
2
6 s + 2 s - 38 = a(s+1)(s+2) + b(s-1)(s+2) + c(s-1)(s+1)
Setting s = 1 yields
6 + 2 - 38 = 6a + 0 + 0
Setting s = -1 yields
6 - 2 - 38 =
0 - 2b + 0
Setting s = -2 yields
24 - 4 - 38 =
0 + 0 + 3c
The same values for a, b and c
then follow.