The Method of Partial Fractions


4. The bottom polynomial can be factored into tex2html_wrap_inline545 . Thus we obtain

eqnarray153


Finding the coefficients

Let's consider the third problem:

displaymath533

Let's first multiply both sides by the least common denominator tex2html_wrap_inline557 :

displaymath548

Then we bring the right hand side into "standard form'':

eqnarray179

We need the left side polynomial and the right side polynomial to coincide; but two polynomials coincide if and only if all their coefficients are the same! Thus

displaymath549

implies the following system of linear equations:

eqnarray181

Let's solve this system and we're done! The third equation implies A=-7, using this information in the first equation yields B=7; last not least, solving the second equation for C we obtain:

displaymath550

This is it. We have computed that

displaymath551


Remarks.

(1) The last step is "pure'' matrix algebra, so you can use any method of solving a system of linear equations you know:

(2) Solving systems of linear equations with "pencil and paper'' is quite error-prone! You can check your answer by combining the partial fractions you obtain into the complicated fraction you started out with.

(2) The computations can be quite cumbersome. For instance, rewriting the fourth problem

displaymath565

using Mathematica yields the partial fraction decomposition

displaymath566

displaymath567

You definitely don't want to do this "by hand''!


Try it yourself!

Here are some problems you can and should do "by hand'':
  1. tex2html_wrap_inline593
  2. tex2html_wrap_inline595
  3. tex2html_wrap_inline597
  4. tex2html_wrap_inline599
Click on the problem to see the answer! Click here to continue.


Helmut Knaust
Fri Jul 5 13:54:22 MDT 1996

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour