Q. 28 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 13–46, write the partial fraction decomposition of each rational expression.

$\frac{10x^2+2x}{{(x-1)}^2(x^2+2)}$

Step-by-Step Solution

Answer

The partial fraction decomposition is $\frac{10x^2+2x}{{(x-1)}^2(x^2+2)}=\frac{14}{3(x-1)}+\frac{4}{{(x-1)}^2}+\frac{-14x+4}{3(x^2+2)}$

Step 1. Given Information

Given rational expression is $\frac{10x^2+2x}{{(x-1)}^2(x^2+2)}$.

We have to write the partial fraction decomposition of the given rational expression.

Step 2. Decompose into partial fractions

We decompose the given rational expression as follows:

$\frac{10x^2+2x}{{(x-1)}^2(x^2+2)}=\frac{A}{(x-1)}+\frac{B}{{(x-1)}^2}+\frac{Cx+D}{(x^2+2)}$

We have to find $A,B,C,D$, so multiply both sides with ${(x-1)}^2(x^2-2)$.

$10x^2+2x=A(x-1)(x^2+2)+B(x^2+2)+(Cx+D){(x-1)}^2$   - Equation-2

Step 3. Finding the value of B

Substitute $x=1$in Equation-2

$$\begin{gathered} 10+2=3B \\ \Rightarrow B=\frac{12}{3} \\ \Rightarrow B=4 \end{gathered}$$

Step 4. Finding A,C,D

• Substitute $x=0$ in Equation-2.

         $$\begin{gathered} 0=A(-1)2+B\left(2\right)+D{(-1)}^2 \\ \Rightarrow 2A-2B-D=0 \end{gathered}$$

          Substitute value of B.

         $2A-D=8$    -Equation-3

• Substitute $x=-1$ in Equation-2

        $$\begin{gathered} 10-2=A(-2)\left(3\right)+B\left(3\right)+(-C+D)4 \\ \Rightarrow 6A+4C-4D+8=3B \end{gathered}$$

          Substitute value of B.

           $$\begin{gathered} 6A+4C-4D+8=12 \\ \Rightarrow 6A+4C-4D=4 \end{gathered}$$

          $3A+2C-2D=2$   - Equation-4

• Substitute $x=2$ in Equation-2

           $$\begin{gathered} 10\left(4\right)+2\left(2\right)=6A+6B+C+D \\ \Rightarrow 6A+C+D-44=-6B \end{gathered}$$

              Substitute value of B.

             $$\begin{gathered} 6A+C+D-44=-24 \\ \Rightarrow 6A+C+D=20 \end{gathered}$$

             $6A+C+D=20$     -Equation-5

Step 5. Solving the system of equations

Solving the equations 3,4,5 will give:'

$$\begin{gathered} A=\frac{14}{3} \\ C=-\frac{14}{3} \\ D=\frac{4}{3} \end{gathered}$$

Therefore by Substituting the value of $A,B,C,D$ the partial fraction decomposition is:

$\frac{10x^2+2x}{{(x-1)}^2(x^2+2)}=\frac{14}{3(x-1)}+\frac{4}{{(x-1)}^2}+\frac{-14x+4}{3(x^2+2)}$