Find the partial fraction expansion of&nbsp;5s3(s2+4)(s2+2s+2)

Step 1 The given fraction is, 5s3(s2+4)(s2+2s+2) The denominator of the fraction has two factors (s2+4) and (s2+2s+2). Since the quadratic equations (s2+4) and (s2+2s+2) are not reducible, the partial fractions must have the form 5s3(s2+4)(s2+2s+2)=As+Bs2+4+Cs+Ds2+2s+2 Step 2 Simplify further as follows. 5s3(s2+4)(s2+2s+2)=As+Bs2+4+Cs+Ds2+2s+2 5s3=(As+B)(s2+2s+2)+(Cs+D)(s2+4) 5s3=s3(B+D)+s2(A+2B+C)+s(2A+2B+4D)+(2A+4C) Equate the coefficients on both sides as follows. B+D=5 A+2B+C=0 2A+2B+4D=0 2A+4C=0 Step 3 On solving the above equations, A=2,B=−8,C=3 andD=4. Substitute A=2,B=−8,C=3 and D=4 in (1) as follows. 5s3(s2+4)(s2+2s+2)=2s+(−8)s2+4+3s+4s2+2s+2

"Find the partial fraction expansion of5s3(s2+4)(s2+2s+2)" was answered by our experts

Ronnie Baur

Answered question

2021-11-17

Find the partial fraction expansion of $\frac{5 s^{3}}{(s^{2} + 4) (s^{2} + 2 s + 2)}$

Answer & Explanation

Parminquale

Beginner2021-11-18Added 17 answers

Step 1
The given fraction is,

\frac{5 s^{3}}{(s^{2} + 4) (s^{2} + 2 s + 2)}

The denominator of the fraction has two factors

(s^{2} + 4) and (s^{2} + 2 s + 2)

.
Since the quadratic equations

(s^{2} + 4) and (s^{2} + 2 s + 2)

are not reducible, the partial fractions must have the form

\frac{5 s^{3}}{(s^{2} + 4) (s^{2} + 2 s + 2)} = \frac{A s + B}{s^{2} + 4} + \frac{C s + D}{s^{2} + 2 s + 2}

Step 2
Simplify further as follows.

\frac{5 s^{3}}{(s^{2} + 4) (s^{2} + 2 s + 2)} = \frac{A s + B}{s^{2} + 4} + \frac{C s + D}{s^{2} + 2 s + 2}

5 s^{3} = (A s + B) (s^{2} + 2 s + 2) + (C s + D) (s^{2} + 4)

5 s^{3} = s^{3} (B + D) + s^{2} (A + 2 B + C) + s (2 A + 2 B + 4 D) + (2 A + 4 C)

Equate the coefficients on both sides as follows.

B + D = 5

A + 2 B + C = 0

2 A + 2 B + 4 D = 0

2 A + 4 C = 0

Step 3
On solving the above equations,

A = 2, B = - 8, C = 3 and D = 4

.
Substitute

A = 2, B = - 8, C = 3 and D = 4

in (1) as follows.

\frac{5 s^{3}}{(s^{2} + 4) (s^{2} + 2 s + 2)} = \frac{2 s + (- 8)}{s^{2} + 4} + \frac{3 s + 4}{s^{2} + 2 s + 2}

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