Chapter 10.7, Problem 21E

To determine

The partial decomposition of the given rational function is to be estimated:

  $\frac{x+14}{x^2-2x-8}$

Answer to Problem 21E

The partial decomposition of $\frac{x+14}{x^2-2x-8}$ is to be estimated as: $\frac{3}{x-4}+\frac{-2}{x+2}$

Explanation of Solution

Given:

  $\frac{x+14}{x^2-2x-8}$

Concept Used:

The process of starting the given rational number with simplified answer steps is the partial-fraction decomposition.

Calculation:

The form of the partial decomposition is expressed by the following expression:

  $\begin{array}{l}\frac{x+14}{x^2-2x-8}=\frac{x+14}{\left(x-4\right)\left(x+2\right)}\\ \frac{x+14}{\left(x-4\right)\left(x+2\right)}=\frac{A}{x-4}+\frac{B}{x+2}\end{array}$

Now, write the right hand side of the equation into a single fraction:

  $\frac{x+14}{\left(x-4\right)\left(x+2\right)}=\frac{\left(x-4\right)B+\left(x+2\right)A}{\left(x-4\right)\left(x+2\right)}$

As the denominators of both eqns. are equal so use the equality of numerators:

  $\begin{array}{l}x+14=\left(x-4\right)B+\left(x+2\right)A\\ x+14=xA+2A+xB-4B\\ x+14=x\left(A+B\right)+2A-4B\end{array}$

The coefficients near the like terms should be equal, so the following system is obtained:

  $\begin{array}{l}A+B=1\mathrm{...}\left(1\right)\\ 2A-4B=14\mathrm{...}\left(2\right)\end{array}$

Eq. (1) cam rewrite as multiply by $2$ :

  $\begin{array}{l}2\left(A+B\right)=2\\ 2A+2B=2\mathrm{...}(3)\end{array}$

Now solve the eq. to get values of $A$ and $B$ by substraction of both eqns(2)and (3).:

  $\begin{array}{l}2A-4B-2A-2B=14-2\\ -6B=12\\ B=-2\end{array}$

Now to get value of $A$ , put the value of $B=-2$ in eq. (1):

  $\begin{array}{l}A+B=1\\ A+-2=1\\ A=3\end{array}$

Therefore, the partial decomposition of the given rational function is to be estimated as:

  $\begin{array}{l}\frac{x+14}{\left(x-4\right)\left(x+2\right)}=\frac{A}{x-4}+\frac{B}{x+2}\\ \frac{x+14}{\left(x-4\right)\left(x+2\right)}=\frac{3}{x-4}+\frac{-2}{x+2}\\ \frac{x+14}{x^2-2x-8}=\frac{3}{x-4}+\frac{-2}{x+2}\end{array}$

Conclusion:

Hence, $\frac{x+14}{x^2-2x-8}=\frac{3}{x-4}+\frac{-2}{x+2}$