Algebra - Partial Fractions

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Section 5.5 : Partial Fractions

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

$\frac{{10x + 35}}{{{{\left( {x + 4} \right)}^2}}}$

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The first step is to determine the form of the partial fraction decomposition. For this problem the partial fraction decomposition is,

$\frac{{10x + 35}}{{{{\left( {x + 4} \right)}^2}}} = \frac{A}{{x + 4}} + \frac{B}{{{{\left( {x + 4} \right)}^2}}}$ Show Step 2

The LCD for this expression is ${\left( {x + 4} \right)^2}$ . Adding the terms back up gives,

$\frac{{10x + 35}}{{{{\left( {x + 4} \right)}^2}}} = \frac{{A\left( {x + 4} \right) + B}}{{{{\left( {x + 4} \right)}^2}}}$ Show Step 3

Setting the numerators equal gives,

$10x + 35 = A\left( {x + 4} \right) + B$ Show Step 4

For this problem we can pick a “good” value of $x$ to determine only one of the constants. Here is that work.

$x = - 4: \hspace{0.25in} - 5 = B \hspace{0.25in} \to \hspace{0.25in}B = - 5$ Show Step 5

To get the remaining constant we can use any value of $x$ and plug that along with the value of $B$ we found in the previous step into the equation from Step 3.

It really doesn’t matter what value of $x$ we pick as long as it isn’t $x = - 4$ since we used that in the previous step. The idea here is to pick a value of $x$ that won’t create “large” or “messy” numbers, if possible. Good choices are often $x = 0$ or $x = 1$ , provided they weren’t used in the previous step of course.

For this problem $x = 0$ seems to be a good choice. Here is the work for this step.

$\begin{align*}10\left( 0 \right) + 35 & = A\left( {0 + 4} \right) + \left( { - 5} \right)\\ 35 & = 4A - 5\\ 40 & = 4A\hspace{0.25in} \Rightarrow \hspace{0.25in}A = 10\end{align*}$ Show Step 6

The partial fraction decomposition is then,

$\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{10x + 35}}{{{{\left( {x + 4} \right)}^2}}} = \frac{{10}}{{x + 4}} - \frac{5}{{{{\left( {x + 4} \right)}^2}}}}}$