Chapter 11, Problem 60RE

To determine

Write the partial fraction decomposition of each rational expression.

Answer to Problem 60RE

  $\boxed{\frac{x}{(x+2)(x-3)}=\frac{2}{x+2}-\frac{3}{x-3}}$

Explanation of Solution

Given information:

Write the partial fraction decomposition of each rational expression.

  $\frac{x}{(x+2)(x-3)}$

Calculation:

The expression to be decomposed into partial fractions is given by the expression given below.

  $y=\frac{x}{(x+2)(x-3)}\mathrm{....}(1)$

The denominator of the given expression is already factorised, so we are not required to factorise the denominator. We decompose the rational expression according to the equation (1).

  $\frac{x}{(x+2)(x-3)}=\frac{A}{x+2}+\frac{B}{x+3}\mathrm{......}(2)$

The value of $A$ and $B$ are to be determined in order to obtain the decomposed form of the given expression.

  $\begin{array}{l}x=A(x-3)+B(x+2)\\ =(A+B)x-3A+2B\end{array}$ ..... (3)

The equation (3) is an identify in $x$ . We equate the coefficients of like powers of $x$ .

  $\{\begin{array}{l}1=A+B\\ 0=-3A+2B\end{array}$

Hence, the values of $A=2$ and $B=3$ . The decomposed form of the given expression is given by the expression given below.

  $\boxed{\frac{x}{(x+2)(x-3)}=\frac{2}{x+2}-\frac{3}{x-3}}$