text: # Partial Fraction Expansion
  
  0 votes
  
  asked Dec 26, 2017
  
  ## 1 Answer
  
  0 votes
  
  The expression
  
  [s^2 + s - 1] / [s(s + 1)] + 1/s
  
  Consider 1st term
  
  [s^2 + s - 1] / [s(s + 1)^2]   =   A/s + B/(s + 1) + C/(s + 1)^2  ------------> (T)
  
  Take the LCM
  
  [s^2 + s - 1] / [s(s + 1)^2]   =   { A[s + 1)^2] + B[s(s + 1)] + Cs } / [ s(s + 1)^2 ]
  
  [s^2 + s - 1]   =   { A[s^2 + 2s + 1)] + B(s^2 + s) + Cs }
  
  [s^2 + s - 1]   =    As^2 + 2As + A + Bs^2 + Bs + Cs
  
  [s^2 + s - 1]   =    (A + B)s^2 + (2A + B + C)s + A
  
  Compare the coefficients of S^2, s and constant terms
  
  A   =   - 1
  
  A + B   =   1  -----------------------------> (1)
  
  2A + B + C   =   1  --------------------------> (2)
  
  Substitute A  = - 1 in equation (1)
  
  -1 + B   =   1
  
  B   =   1 + 1
  
  B   =   2
  
  Substitute A  = - 1 and B  =  2 in equation (2)
  
  2(-1) + 2 + C   =   1
  
  -2 + 2 + C   =   1
  
  C   =   1
  
  Substitute A  = - 1, B  =  2 and C  =  1 in equation (T)
  
  [s^2 + s - 1] / [s(s + 1)^2]   =   (-1/s) + 2/(s + 1) + 1/(s + 1)^2
  
  Hence
  
  [s^2 + s - 1] / [s(s + 1)] + 1/s   =   [ (-1/s) + 2/(s + 1) + 1/(s + 1)^2 ] + 1/s
  
  [s^2 + s - 1] / [s(s + 1)] + 1/s   =    2/(s + 1) + 1/(s + 1)^2
  
  answered Dec 27, 2017
explanation: The document contains a detailed solution to a partial fraction expansion problem. It starts with the expression \(\frac{s^2 + s - 1}{s(s + 1)} + \frac{1}{s}\) and proceeds to break it down step-by-step, ultimately arriving at the simplified form \(\frac{2}{s + 1} + \frac{1}{(s + 1)^2}\).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Partial Fraction Expansion
  
  0 votes
  
  asked Dec 26, 2017
  
  ## 1 Answer
  
  0 votes
  
  The expression
  
  [s^2 + s - 1] / [s(s + 1)] + 1/s
  
  Consider 1st term
  
  [s^2 + s - 1] / [s(s + 1)^2]   =   A/s + B/(s + 1) + C/(s + 1)^2  ------------> (T)
  
  Take the LCM
  
  [s^2 + s - 1] / [s(s + 1)^2]   =   { A[s + 1)^2] + B[s(s + 1)] + Cs } / [ s(s + 1)^2 ]
  
  [s^2 + s - 1]   =   { A[s^2 + 2s + 1)] + B(s^2 + s) + Cs }
  
  [s^2 + s - 1]   =    As^2 + 2As + A + Bs^2 + Bs + Cs
  
  [s^2 + s - 1]   =    (A + B)s^2 + (2A + B + C)s + A
  
  Compare the coefficients of S^2, s and constant terms
  
  A   =   - 1
  
  A + B   =   1  -----------------------------> (1)
  
  2A + B + C   =   1  --------------------------> (2)
  
  Substitute A  = - 1 in equation (1)
  
  -1 + B   =   1
  
  B   =   1 + 1
  
  B   =   2
  
  Substitute A  = - 1 and B  =  2 in equation (2)
  
  2(-1) + 2 + C   =   1
  
  -2 + 2 + C   =   1
  
  C   =   1
  
  Substitute A  = - 1, B  =  2 and C  =  1 in equation (T)
  
  [s^2 + s - 1] / [s(s + 1)^2]   =   (-1/s) + 2/(s + 1) + 1/(s + 1)^2
  
  Hence
  
  [s^2 + s - 1] / [s(s + 1)] + 1/s   =   [ (-1/s) + 2/(s + 1) + 1/(s + 1)^2 ] + 1/s
  
  [s^2 + s - 1] / [s(s + 1)] + 1/s   =    2/(s + 1) + 1/(s + 1)^2
  
  answered Dec 27, 2017
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a detailed solution to a partial fraction expansion problem. It starts with the expression \(\frac{s^2 + s - 1}{s(s + 1)} + \frac{1}{s}\) and proceeds to break it down step-by-step, ultimately arriving at the simplified form \(\frac{2}{s + 1} + \frac{1}{(s + 1)^2}\).
  
  Label: question-and-answer