Chapter G, Problem 1E

(a)

To determine

To calculate:

The partial fraction decomposition equation for the following function:

  $\frac{2x}{\left(x+3\right)\left(3x+1\right)}$

Answer to Problem 1E

The partial fraction decomposition equation of $\frac{2x}{\left(x+3\right)\left(3x+1\right)}$ is

  $\frac{2x}{\left(x+3\right)\left(3x+1\right)}=\frac{A}{\left(3x+1\right)}+\frac{B}{\left(x+3\right)}$ .

Explanation of Solution

Given information:

  $\frac{2x}{\left(x+3\right)\left(3x+1\right)}$

Calculation:

The equation is:

  $\frac{2x}{\left(x+3\right)\left(3x+1\right)}$

Need to convert the above equation in the form of partial fraction decomposition:

  $\frac{2x}{\left(x+3\right)\left(3x+1\right)}=\frac{A}{\left(3x+1\right)}+\frac{B}{\left(x+3\right)}$

(b)

To determine

To calculate:

The partial fraction decomposition equation for the following function:

  $\frac{1}{x^3+2x^2+x}$

Answer to Problem 1E

The partial fraction decomposition equation of $\frac{1}{x^3+2x^2+x}$ is

  $\frac{A}{x}+\frac{B}{\left(x+1\right)}+\frac{C}{{\left(x+1\right)}^2}$ .

Explanation of Solution

Given information:

  $\frac{1}{x^3+2x^2+x}$

Calculation:

The equation is:

  $\frac{1}{x^3+2x^2+x}$

First simplify the given equations as:

  $\begin{array}{l}\frac{1}{x^3+2x^2+x}=\frac{1}{x\left(x^2+2x+1\right)}\\ \frac{1}{x^3+2x^2+x}=\frac{1}{x{\left(x+1\right)}^2}\end{array}$

Need to convert the above equation in the form of partial fraction decomposition:

  $\frac{1}{x{\left(x+1\right)}^2}=\frac{A}{x}+\frac{B}{\left(x+1\right)}+\frac{C}{{\left(x+1\right)}^2}$