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Express \(\cfrac{3s - 1}{(s + 1)(s - 3)}\) as partial fractions


Question

Express \(\cfrac{3s - 1}{(s + 1)(s - 3)}\) as partial fractions

Options

A) \(\frac{1}{(s+1)}+\frac{2}{(s-3)}\)

B) \(\frac{2}{(s+1)}-\frac{1}{(s-3)}\)

C) \(\frac{2}{(s+3)}+\frac{1}{(s-1)}\)

D) \(\frac{1}{(s-1)}+\frac{2}{(s-3)}\)

The correct answer is A.

Explanation:

3s-1\(s+1)(s-3)=A/(s+1)+B/(s-3)
3s-1\{s+1)(s-3)=A(s-3)+B(s+1)
3s-1/(s+1)(s-3)=A(s-3)+B(s+1)/(s+1)(s-3)
denominator cancels each other
3s-1=A(s-3)+B(s+1)
If s=+3
3(3)-1=A(3-3)+B(3+1)
[A(0)=0]
9-1=4B
8=4B
B=8/4
B=2✓
If s=-1
3s-1=A(s-3)+B(s+1)
3(-1)-1=A(-1-3)+B(-1+1)
-4=-4A
A=-4/-4
A=1✓
Therefore,1/(s+1)+2/(s-3) is the answer

Explanation provided by Franklin Precious


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Discussion (2)

  • Do trial and error...the one that gives you back the question is the answer

  • 3s-1\(s+1)(s-3)=A/(s+1)+B/(s-3)
    3s-1\{s+1)(s-3)=A(s-3)+B(s+1)
    3s-1/(s+1)(s-3)=A(s-3)+B(s+1)/(s+1)(s-3)
    denominator cancels each other
    3s-1=A(s-3)+B(s+1)
    If s=+3
    3(3)-1=A(3-3)+B(3+1)
    [A(0)=0]
    9-1=4B
    8=4B
    B=8/4
    B=2✓
    If s=-1
    3s-1=A(s-3)+B(s+1)
    3(-1)-1=A(-1-3)+B(-1+1)
    -4=-4A
    A=-4/-4
    A=1✓
    Therefore,1/(s+1)+2/(s-3) is the answer