Chapter 7.4, Problem 50E

To determine

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

Answer to Problem 50E

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

Explanation of Solution

Given information:

The improper rational expression is $\frac{2x^4+8x^3+7x^2-7x-12}{x^3+4x^2+4x}$ .

Calculations:

Since the given rational expression is improper, lets begin by dividing the numerator by the denominator.

Therefore, the long division result is

  $2x+\frac{-x^2-7x-12}{x^3+4x^2+4x}$

The denominator of the remainder can be factored so the result is

  $\frac{-x^2-7x-12}{x{(x+2)}^2}$

Now, continuing the partial fraction on remainder, the decomposition form is as follows:

  $\frac{-x^2-7x-12}{x{(x+2)}^2}=\frac{A}{x}+\frac{B}{x+2}+\frac{C}{{(x+2)}^2}$

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

That is,

  $\begin{array}{l}$ -x^2-7x-12=A{(x+2)}^2+Bx(x+2)+Cx$\Rightarrow -x^2-7x-12=A(x^2+4x+4)+B(x^2+2x)+Cx\\ \Rightarrow -x^2-7x-12=A{x}^2+4Ax+4A+B{x}^2+2Bx+Cx\\ \Rightarrow -x^2-7x-12=(A+B)x^2+(4A+2B+C)x+(4A)\end{array}$

Comparing the coefficients of like terms, we get

  $\{\begin{array}{c}A+B=-1\\ 4A+2B+C=-7\\ 4A=-12\end{array}$

Using $4A=-12$ , we get $A=-3.$

Then,

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

And

  $\begin{array}{l}4A+2B+C=-7\\ \Rightarrow 4(-3)+2(2)+C=-7\\ \Rightarrow -12+4+C=-7\\ \Rightarrow C=1\end{array}$

Conclusion:

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