Chapter 7.3, Problem 74E

To determine

To calculate:

The partial fraction decomposition for the given rational expression $\frac{2x^2+x+8}{x^4+8x^2+16}$ and check the result graphically.

Answer to Problem 74E

  $\frac{2}{x^2+4}+\frac{x}{{\left(x^2+4\right)}^2}$

Explanation of Solution

Given information: The given rational expression $\frac{2x^2+x+8}{x^4+8x^2+16}$

Calculation:

Consider $\frac{2x^2+x+8}{x^4+8x^2+16}$ ,

Factorize $x^4+8x^2+16\Rightarrow {\left(x^2+4\right)}^2$

  $=\frac{2x^2+x+8}{{\left(x^2+4\right)}^2}$

Create the partial fraction template using the template ${\left(x^2+4\right)}^2$

  $\frac{2x^2+x+8}{{\left(x^2+4\right)}^2}=\frac{a_{1}x+a_0}{x^2+4}+\frac{a_{3}x+a_2}{{\left(x^2+4\right)}^2}$

Multiply equation by the denominator

  $\frac{\left(2x^2+x+8\right){\left(x^2+4\right)}^2}{{\left(x^2+4\right)}^2}=\frac{\left(a_{1}x+a_0\right){\left(x^2+4\right)}^2}{x^2+4}+\frac{\left(a_{3}x+a_2\right){\left(x^2+4\right)}^2}{{\left(x^2+4\right)}^2}$

Simplify,

  $2x^2+x+8=\left(a_{1}x+a_0\right)\left(x^2+4\right)+a_{3}x+a_2$

Expand,

  $2x^2+x+8=a_1{x}^3+4a_{1}x+a_0{x}^2+4a_0+a_{3}x+a_2$

Group the elements according to powers of $x$ ,

  $2x^2+1\cdot x+8=a_1{x}^3+a_0{x}^2+x\left(4a_1+a_3\right)+\left(4a_0+a_2\right)$

Equate the co-efficient of similar terms on both sides,

  $\begin{array}{l}4a_0+a_2=8\\ 4a_1+a_3=1\\ a_0=2\\ a_1=0\end{array}$

Solve the system of equations:

  $a_0=2,a_2=0,a_1=0,a_3=1$

Substitute the values in the partial fraction,

  $\frac{0\cdot x+2}{x^2+4}+\frac{1\cdot x+0}{{\left(x^2+4\right)}^2}$

Simplify,

  $\frac{2}{x^2+4}+\frac{x}{{\left(x^2+4\right)}^2}$

Hence, the solution of partial fraction decomposition for the given rational expression

  $\frac{2x^2+x+8}{x^4+8x^2+16}=\frac{2}{x^2+4}+\frac{x}{{\left(x^2+4\right)}^2}$

Graph:

  

  

Interpretation:

The graph of the partial fraction is same as the solution we solved.