Chapter 7.4, Problem 54E

To determine

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

Answer to Problem 54E

The partial fraction decomposition of the given rational expression is $\frac{3}{x}+\frac{-3x+3}{x^2+2}$ .

Explanation of Solution

Given information:

The rational expression is $\frac{3x+6}{x^3+2x}$ .

Calculations:

The factored form of the denominator $x^3+2x$ is $x(x^2+2)$ .

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

  $\frac{3x+6}{x(x^2+2)}=\frac{A}{x}+\frac{Bx+C}{x^2+2}$

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

That is,

  $\begin{array}{l}3x+6=A(x^2+2)+(Bx+C)x\\ \Rightarrow 3x+6=A{x}^2+2A+B{x}^2+Cx\\ \Rightarrow 3x+6=(A+B)x^2+Cx+2A\end{array}$

Comparing the coefficients of like terms, we get

  $\{\begin{array}{c}A+B=0\\ C=3\\ 2A=6\end{array}$

We got $C=3$ .

Solving the third equation, we get

  $\begin{array}{l}2A=6\\ \Rightarrow A=3\end{array}$

Substituting the value in the first equation, we get

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

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

Conclusion:

Using Desmos graphing utility, we can observe that both curves are overlapping each other.