Section 5.5 : Partial Fractions
2. Determine the partial fraction decomposition of each of the following expression.
\[\frac{{34 - 12x}}{{3{x^2} - 10x - 8}}\]Show All Steps Hide All Steps
Start SolutionThe 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,
\[\frac{{34 - 12x}}{{\left( {3x + 2} \right)\left( {x - 4} \right)}}\]Okay, we can now see that the partial fraction decomposition is,
\[\frac{{34 - 12x}}{{3{x^2} - 10x - 8}} = \frac{A}{{3x + 2}} + \frac{B}{{x - 4}}\] Show Step 2The LCD for this expression is \(\left( {3x + 2} \right)\left( {x - 4} \right)\). Adding the two terms back up gives,
\[\frac{{34 - 12x}}{{3{x^2} - 10x - 8}} = \frac{{A\left( {x - 4} \right) + B\left( {3x + 2} \right)}}{{\left( {3x + 2} \right)\left( {x - 4} \right)}}\] Show Step 3Setting the numerators equal gives,
\[34 - 12x = A\left( {x - 4} \right) + B\left( {3x + 2} \right)\] Show Step 4Now all we need to do is pick “good” values of \(x\) to determine the constants. Here is that work.
\[\begin{array}{l}{x = 4:}\\{\displaystyle x = - \frac{2}{3}:}\end{array}\hspace{0.25in}\begin{aligned} - 14 & = 14B\\42 & = - \frac{{14}}{3}A\end{aligned}\hspace{0.25in} \to \hspace{0.25in}\begin{array}{l}{A = - 9}\\{B = - 1}\end{array}\] Show Step 5The partial fraction decomposition is then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{34 - 12x}}{{3{x^2} - 10x - 8}} = - \frac{9}{{3x + 2}} - \frac{1}{{x - 4}}}}\]