Calculus Methods

03 Separating Partial Fractions

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        1) Completely factor the denominator, and

        completely expand the numerator. Each factor in

        the denominator should now be either linear or

        quadratic.

 

            2) Write an equation with the new fraction on one

        side; on the other side, write a sum of fractions:

        each fraction will use a constant in the numerator

        and a separate factor (from the new fraction in

        part 1) as its denominator. If a factor appears more

        than once, (for instance in ), then you

        will need to use two separate fraction, one for each

        power in the factor (for instance .

        Also, if the factor in the denominator is quadratic,

        then the numerator must be linear.

 

        3) On one side of the equation, you now have a sum

        of many fractions. Combine those fractions into a

        single fraction and completely expand the

        numerator.

 

        4) Since the fractions obtained in steps 1 and 3 are

        equal, their numerators must be equal. Also, since

        there are variables on each side, the coefficients of

        each individual power on each side must be equal.

        That means that you can set up a series of

        equations. If done properly, you will have the same

        number of equations as unknowns.

 

        5) Solve the equations to get the unknowns.

 

        6) Write out the fractions.

 

 

 

        Example #1: Separate the following fraction into

        partial fractions:

 

 

        1) The numerator is already expanded, and the

        denominator needs to be factored.

                             

 

 

        2) The factor occurs twice, so we will need to

        separate fractions for that factor.

                 

                 

 

 

        3)    

                 

 

 

 

        4)    

                 

        

                  Setting up the three equations:

                          

 

        

        5)     Now use your favorite method of solving a

        system of equations:

                          

 

 

        6)            

 

 

 

 

 

 

        Example #2: Separate the following fraction into

        partial fractions:

        

 

        1) The numerator needs to be expanded and the

        denominator needs to be factored.

                 

 

 

        2) One factor () occurs twice, so we will need to

        fractions for . Also, we have a quadratic factor, so

        its numerator will be linear.

 

       

 

        Note that we have six unknowns, so we should end

        up with six equations in step 4.

 

 

        3)     

            

        

 

        4)

             

 

                  Setting up the 6 equations:

                          

 

 

 

        5)     Now use your favorite method of solving a

        system of equations:           

                 

 

 

        6)

 

 

On to Method 04 - Finding Limits

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