Solution:
We beginby factoringthe denominatoras Thenwritethe partialfraction
decompositionas
Our goal at this pointis to find the valuesof A and B. To solvethis equation,multiplyboth sidesof the
equationby the factoreddenominator This processwill producethebasicequation.
This equationis true for all valuesof The mostconvenientvaluesare the onesthat makea factorequal
to zero,namely, and Substituting
Similarly, substitutingfor into the basicequationwe get
We havesolvedthe basicequationby findingthe valuesof and Therefore,the partialfractiondecom-
positionis
GeneralDescriptionof the Method
To be able to writea rationalfunction as a sum of partialfractions,mustapplytwo conditions:
- The degreeof mustbe less than the degreeof If so, the rationalfunctionis calledproper.
If it is not, divide by (use long division)and workwith the remainderterm. - The factorsof are known.If not, you needto find a way to find them.The guidebelowshowshow
you can write as a sum of partialfractionsif the factorsof are known.
A Guideto FindingPartialFractionsDecompositionof a RationalFunction