Chapter 7.3, Problem 73E

To determine

To Find : Partial fraction of rational expression $\frac{x}{x^3-x^2-2x+2}$

Answer to Problem 73E

The partial fraction of expansion is $\frac{-1}{x-1}+\frac{x+2}{x^2-2}$

Explanation of Solution

Given information:

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

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

Here the denominator contain linear and quadratic term as factors and the given rational expression can be expressed as sum of partial fraction.

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

Now , for finding the value of coefficients $A$ and $B$

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

Comparing coefficients of $x^2$

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

Comparing coefficients of $x$

  $\begin{array}{l}C-B=1\\ C+A=1\\ C=1-A\end{array}$

Comparing constant terms in the equation

  $\begin{array}{l}-2A-C=0\\ -2A-1+A=0\\ A=-1\\ C=2\\ B=1\end{array}$

therefore ,

Checking algebraically,

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

Checking Graphically,

  $\frac{x}{x^3-x^2-2x+2}$

  

  $\frac{-1}{x-1}+\frac{x+2}{x^2-2}$

  

Interpretation:

The graph of rational expression and partial fraction is same hence graphically it is proved that $\frac{x}{x^3-x^2-2x+2}=\frac{-1}{x-1}+\frac{x+2}{x^2-2}$