Chapter 7, Section 5

Partial fractions



Decomposition of N(x) / D(x) into Partial Fractions

1.  Divide if improper:  If N(x)/D(x) is an improper fraction ( that is, id the degree of the numerator is greater than or equal to the degree of the denominator), divide the denominator into the numerator to obtain

N(x)/D(x)  = ( a polynomial) + N1(x)/D(x) 

2. Factor Denominator: Completely factor the denominator into factors of the form

   ( px  + q )m    and  ( ax +  bx  +  c )n

where ax2 + bx + c   is irreducible.

3.  Linear factors:  For each factor of the form ( px  + q )m , the partial fraction decomposition must include the following sum of m fractions.  

  (A1/( px  + q ))  +  (A2/( px  + q )2)  + ... .+  (Am/( px  + q )m)

4.  Quadratic factors: For each factor of the form ( ax +  bx  +  c )n, the partial fraction decomposition must include the following sum of n fractions.

 ((B1x + C1)/( ax +  bx  +  c ))  +  ((B2x + C2)/(ax +  bx  +  c )2)  + ... .+  ((Bnx + Cn)/( ax +  bx  +  c )n)



Guidelines for solving the basic Equations

Linear Fractions

1.  Substitute the roots of the distinct linear factors into the basic equation.

2.  For repeated linear factors, use the coefficients determined in guideline 1 to rewrite the basic equations.  Then substitute other convenient values of x and solve the remaining coefficients.



Quadratic Fractions

1.  Expand the basic equation.

2.  Collect terms according to powers of x.

3.  Equate the coefficients of  like powers to obtain a system of linear equations involving A,B,C, and so on.

4.  Solve the system of linear equations.





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