# Method of Partial Fractions

Jun 21st, 2015
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Method of Partial FractionTo decompose the rational function :Step 1 Use long division (if necessary) to ensure that the degree of is less than the degree of (see Breaking up a rational function in section1.1).Step 2 Factor Q(x) as far as possible.Step 3 Write down the correct form for the partial fraction decomposition (see below) and solve for the constants.To factor Q(x) we have to write it as a product of linear factors (of the form ) and irreducible quadratic factors (of the form with ).Some of the factors could be repeated. For instance if we factor asIt is important that in each quadratic factor we have , otherwise it is possible to factor that quadratic piece further. For example if then we can writeWe will now show how to write as a sum of terms of the formand Exactly how to do this depends on the factorization of and we now give four cases that can occur.Q(x) is a product of linear factors with no repeatsThis means that where no factor is repeated and no factor is a multiple of another.For each linear term we write down something of the form , so in total we writeExample 1Find Here we have and Q(x) is a product of linear factors. So we writeMultiply both sides by the denominatorSubstitute in three values of x to get three equations for the unknown constants,so , andWe can now integrate the left hand side.ExercisesEvaluate the following by the method partial fraction decomposition.1. 2. SolutionsQ(x) is a product of l

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