Chapter 7.3, Problem 76E

To determine

To calculate:

The partial fraction decomposition for the given rational expression $y=\frac{2\left(4x-3\right)}{x^2-9}$ and check the result graphically.

Answer to Problem 76E

  $\frac{3}{x-3}+\frac{5}{x+3}$

Explanation of Solution

Given information:

The given rational expression $y=\frac{2\left(4x-3\right)}{x^2-9}$

Calculation:

Consider $y=\frac{2\left(4x-3\right)}{x^2-9}$

  $y=\frac{2\left(4x-3\right)}{x^2-9}=\frac{8x-6}{x^2-9}=\frac{a}{x-3}+\frac{b}{x+3}=\frac{a\left(x+3\right)+b\left(x-3\right)}{x^2-9}$

Simplifying and equate the co-efficient of similar terms on both sides, $\begin{array}{l}\left(a+b\right)x+\left(3a-3b\right)=8x-6\\ a+b=8\\ 3a-3b=-6\end{array}$

  $\begin{array}{l}\left[\begin{array}{cc}1& 1\\ 3& -3\end{array}|\begin{array}{c}8\\ -6\end{array}\right]\Rightarrow \left[\begin{array}{cc}1& 1\\ 1& -1\end{array}|\begin{array}{c}8\\ -2\end{array}\right]\\ \left[\begin{array}{cc}1& 1\\ 0& -2\end{array}|\begin{array}{c}8\\ -10\end{array}\right]\to \left[\begin{array}{cc}1& 1\\ 0& 1\end{array}|\begin{array}{c}8\\ 5\end{array}\right]\\ \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}|\begin{array}{c}3\\ 5\end{array}\right]\end{array}$

  $a=3;b=5$

Therefore the solution is,

  $y=\frac{8x-6}{x^2-9}=\frac{3}{x-3}+\frac{5}{x+3}$

Graph:

  

Interpretation:

The graph with purple colour is the term with denominator $x-3$ has $x=3$ as a vertical asymptote.

The graph with teal colour is the term with denominator $x+3$ has $x=-3$ as a vertical asymptote.

The graph of the function is the black graph and it has both vertical asymptotes.