Chapter 7.4, Problem 41E

To determine

To find:The partial fraction decomposition of the given proper rational expression.

Answer to Problem 41E

The partial fraction decomposition of the given rational expressionis $\frac{2}{x}-\frac{3}{x^2}+\frac{-2x+3}{x^2+2}+\frac{-4x+6}{{(x^2+2)}^2}$ .

Explanation of Solution

Given information:

The proper rational expression is $\frac{8x-12}{x^2{(x^2+2)}^2}$ .

Calculations:

Based on the factors of the denominator, the decomposition form is as follows:

  $\frac{8x-12}{x^2{(x^2+2)}^2}=\frac{A}{x}+\frac{B}{x^2}+\frac{Cx+D}{x^2+2}+\frac{Ex+F}{{(x^2+2)}^2}$

Now, multiply each side by the least common multiple, $x^2{(x^2+2)}^2$ , to obtain the basic equation.

That is,

  $\begin{array}{l}$ 8x-12=Ax{(x^2+2)}^2+B{(x^2+2)}^2+(Cx+D)x^2(x^2+2)+(Ex+F)x^2$\Rightarrow 8x-12=Ax(x^4+4x^2+4)+B(x^4+4x^2+4)+(Cx+D)(x^4+2x^2)+(Ex+F)x^2\\ \Rightarrow 8x-12=A{x}^5+4A{x}^3+4Ax+B{x}^4+4B{x}^2+4B+C{x}^5+2C{x}^3+D{x}^4+2D{x}^2+E{x}^3+F{x}^2\\ \Rightarrow 8x-12=(A+C)x^5+(B+D)x^4+(4A+2C+E)x^3+(4B+2D+F)x^2+(4A)x+(4B)\end{array}$

Comparing the coefficients of like terms, we get

  $\{\begin{array}{c}A+C=0\\ B+D=0\\ 4A+2C+E=0\\ 4B+2D+F=0\\ 4A=8\\ 4B=-12\end{array}$

Solving the last two equations,

  $\begin{array}{l}4A=8\Rightarrow A=2\\ 4B=-12\Rightarrow B=-3\end{array}$

Using these values,

  $\begin{array}{l}A+C=0\Rightarrow 2+C=0\Rightarrow C=-2\\ B+D=0\Rightarrow -3+D=0\Rightarrow D=3\end{array}$

Now, solving the third and fourth equations,

  $\begin{array}{l}4A+2C+E=0\Rightarrow 4(2)+2(-2)+E=0\Rightarrow 8-4+E=0\Rightarrow E=-4\\ 4B+2D+F=0\Rightarrow 4(-3)+2(3)+F=0\Rightarrow -12+6+F=0\Rightarrow F=6\end{array}$

Conclusion:

Therefore, the partial fraction decomposition of the given rational expression is $\frac{2}{x}-\frac{3}{x^2}+\frac{-2x+3}{x^2+2}+\frac{-4x+6}{{(x^2+2)}^2}$ .