Q. 14 - Precalculus Enhanced with Graphing Utilities

Question

Write the partial fraction decomposition of each rational expression:

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

Step-by-Step Solution

Answer

The partial fraction decomposition is,

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

Step 1. Given Information.

The rational expression is,

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

Step 2. Decomposition of Partial Fractions.

Decompose the rational expressions,

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

Multiplying each side by,

$$\begin{gathered} 3x=A(x-1)+B(x+2) \\ 3x=(A+B)x+(-A+2B)..............\left(2\right) \end{gathered}$$

Equating the coefficients of like powers of $x$ to get,

$$\begin{gathered} A+B=3 \\ -A+2B=0 \end{gathered}$$

Solving the variables,

$$\begin{gathered} A=2 \\ B=1 \end{gathered}$$

The partial fraction decomposition is,

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

Step 3. Checking the solution.

Adding the rational expression,

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

                        $$\begin{gathered} =\frac{2x-2+x+2}{(x+2)(x-1)} \\ =\frac{3x}{(x+2)(x-1)} \end{gathered}$$