Algebra - Partial Fractions

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Section 5.5 : Partial Fractions

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1. Determine the partial fraction decomposition of each of the following expression.

\[\frac{{17x - 53}}{{{x^2} - 2x - 15}}\]

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The first step is to determine the form of the partial fraction decomposition. However, in order to do that we first need to factor the denominator as much as possible. Doing this gives,

\[\frac{{17x - 53}}{{\left( {x - 5} \right)\left( {x + 3} \right)}}\]

Okay, we can now see that the partial fraction decomposition is,

\[\frac{{17x - 53}}{{{x^2} - 2x - 15}} = \frac{A}{{x - 5}} + \frac{B}{{x + 3}}\] Show Step 2

The LCD for this expression is \(\left( {x - 5} \right)\left( {x + 3} \right)\). Adding the two terms back up gives,

\[\frac{{17x - 53}}{{{x^2} - 2x - 15}} = \frac{{A\left( {x + 3} \right) + B\left( {x - 5} \right)}}{{\left( {x - 5} \right)\left( {x + 3} \right)}}\] Show Step 3

Setting the numerators equal gives,

\[17x - 53 = A\left( {x + 3} \right) + B\left( {x - 5} \right)\] Show Step 4

Now all we need to do is pick “good” values of \(x\) to determine the constants. Here is that work.

\[\begin{array}{l}{x = 5:}\\{x = - 3:}\end{array}\hspace{0.25in}\begin{aligned}32 & = 8A\\ - 104 & = - 8B\end{aligned}\hspace{0.25in}\to \hspace{0.25in}\begin{array}{l}{A = 4}\\{B = 13}\end{array}\] Show Step 5

The partial fraction decomposition is then,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{17x - 53}}{{{x^2} - 2x - 15}} = \frac{4}{{x - 5}} + \frac{{13}}{{x + 3}}}}\]