The partial fraction expansion for the given rational function is:
4s/ (s−7)(s+7)(s−7) = 2/s−7 + 2/s+7
To find the partial fraction expansion of the given rational function, we need to express it as the sum of simpler fractions with unknown numerators over the individual factors in the denominator. The denominator has three factors:
(s−7), (s+7), and (s−7).
The partial fraction decomposition will have the form:
4s/ (s−7)(s+7)(s−7) = A/s−7 + B/s+7 + C/(s−7)62
To find A, B, and C, we can use various methods. One common approach is to clear the fractions by multiplying both sides by the denominator of the original expression:
4s=A(s+7)+B(s−7)+C(s−7)^2
Now, we can solve for A, B, and C.
Let's go through the steps:
4s=A(s+7)+B(s−7)+C(s−7)^2
Set s=7 to eliminate B and C:
4(7)=A(7+7)+0+0
28=14A
A=2
Set s=−7 to eliminate A and C:
4(−7)=0+B(−7−7)+0
−28=−14B
B=2
Set s=0 to eliminate A and B:
4(0)=0+0+C(0−7)^2
0=49C
C=0
Now we can write the partial fraction expansion
4s/ (s−7)(s+7)(s−7) = 2/s−7 + 2/s+7
So, the partial fraction expansion for the given rational function is:
4s/ (s−7)(s+7)(s−7) = 2/s−7 + 2/s+7
Question
Determine the partial fraction expansion for the rational function below:
4s / (s-7) (s²-49) =