Question

Write the partial fraction decomposition of each rational expression.

$\frac{3x}{(x+2)(x-4)}$.

Short answer

The partial fraction decomposition is$\frac{1}{x+2}+\frac{2}{x-4}$.

Step-by-step solution

Step 1. Given Information.

The rational expression is,

$\frac{3x}{(x+2)(x-4)}$.

Step 2. Partial Fraction Decomposition.

The decomposition of the partial fraction, we get,

$\frac{3x}{(x+2)(x-4)}=\frac{A}{x+2}+\frac{B}{x-4}............\left(1\right)$.

Multiplying both sides by $(x+2)(x-4)$,

$$\begin{gathered} 3x=A(x-4)+B(x+2) \\ 3x=(A+B)x+(-4A+2B).........\left(2\right) \end{gathered}$$

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

$$\begin{gathered} A+B=3........\left(3\right) \\ -4A+2B=0 \\ \Rightarrow -2A+B=0........\left(4\right) \end{gathered}$$

Solving the equations we get,

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

The partial fraction decomposition is,

$\frac{3x}{(x+2)(x-4)}=\frac{1}{x+2}+\frac{2}{x-4}$.

Step 3. Checking the solution.

Adding the rational expressions,

$\frac{1}{x+2}+\frac{2}{x-4}=\frac{1(x-4)+2(x+2)}{(x+2)(x-4)}$

$$\begin{gathered} =\frac{x-4+2x+4}{(x+2)(x-4)} \\ =\frac{3x}{(x+2)(x-4)} \end{gathered}$$

The solution is true.