Partial Fractions

 

 

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Partial Fractions

 

Linear Factors

 

Repeated Linear Factors

 

Quadratic Factors

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Improper Fractions

 

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Quadratic Factors
 
Here are some examples on finding partial fractions for fractions with quadratic factors.
 
 
  Express in partial fractions:
                
 
  Let
        
 
    Using "cover-up" rule,
                C = (3+2+7) / (1+5)
                   = 2
 
    3x2 + 2x + 7 = (Ax + B) (x - 1) + 2 (x2 + 5)
                      = x2 (A + 2) + x (B - A) + 10 - B
 
    Comparing coefficient of x2 :
                  A + 2 = 3
                        A = 1
 
    Comparing constant term:
                 10 - B = 7
                       B = 3
 
 Therefore,
       
 
 
Express in partial fractions:
                  
 
  Let
        
 
    Using cover-up rule,
                       D = (3 -8 -3 -4) / (1 + 3)
                           = -3
 
     3 - 8x - 3x2 - 4x3 = (Ax + B) (x - 1)2 + C (x2 + 3) (x - 1) - 3 (x2 + 3)
                             = x3 (A + C) + x2 (B - 2A - C - 3) + x (A - 2B + 3C) + B -3C - 9
 
    Comparing coefficient of x3 :
                           A + C = -4
                                 A = -4 - C
 
    Comparing constant term:
                     B - 3C - 9 = 3
                                B = 12 + 3C
 
    Comparing coefficient of x:
                   A - 2B + 3C = -8
  -4 - C - 2(12 + 3C) + 3C = -8               substituting the above equations
                                  C = -5
 
       A = 1, B = -3
 
 Therefore