Chapter 7.4, Problem 42E

To determine

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

Answer to Problem 42E

The partial fraction decomposition of the given rational expressionis $-\frac{2}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\frac{2x-1}{x^2+1}+\frac{x-1}{{(x^2+1)}^2}$ .

Explanation of Solution

Given information:

The proper rational expression is $\frac{x+1}{x^3{(x^2+1)}^2}$ .

Calculations:

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

  $\frac{x+1}{x^3{(x^2+1)}^2}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{Dx+E}{x^2+1}+\frac{Fx+G}{{(x^2+1)}^2}$

Now, multiply each side by the least common multiple, $x^3{(x^2+1)}^2$ , to obtain the basic equation.

That is,

  $\begin{array}{l}$ x+1=A{x}^2{(x^2+1)}^2+Bx{(x^2+1)}^2+C{(x^2+1)}^2+(Dx+E)x^3(x^2+1)+(Fx+G)x^3$\Rightarrow x+1=A{x}^2(x^4+2x^2+1)+Bx(x^4+2x^2+1)+C(x^4+2x^2+1)+(Dx+E)(x^5+x^3)+(Fx+G)x^3\\ \Rightarrow x+1=A{x}^6+2A{x}^4+A{x}^2+B{x}^5+2B{x}^3+Bx+C{x}^4+2C{x}^2+C+D{x}^6+D{x}^4+E{x}^5+E{x}^3+F{x}^4+G{x}^3\\ \Rightarrow x+1=(A+D)x^6+(B+E)x^5+(2A+C+D+F)x^4+(2B+E+G)x^3+(A+2C)x^2+(B)x+(C)\end{array}$

Comparing the coefficients of like terms, we get

  $\{\begin{array}{c}A+D=0\\ B+E=0\\ 2A+C+D+F=0\\ 2B+E+G=0\\ A+2C=0\\ B=1\\ C=1\end{array}$

Using $C=1,$

  $A+2C=0\Rightarrow A+2(1)=0\Rightarrow A=-2$

Solving the first two equations,

  $\begin{array}{l}A+D=0\Rightarrow D=-A\Rightarrow D=2\\ B+E=0\Rightarrow E=-B\Rightarrow E=-1\end{array}$

Using these values, solving the third and fourth equations

  $\begin{array}{l}2A+C+D+F=0\Rightarrow 2(-2)+1+2+F=0\Rightarrow F=1\\ 2B+E+G=0\Rightarrow 2(1)+(-1)+G=0\Rightarrow G=-1\end{array}$

Conclusion:

Therefore, the partial fraction decomposition of the given rational expression is $-\frac{2}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\frac{2x-1}{x^2+1}+\frac{x-1}{{(x^2+1)}^2}$ .