Q. 34 - Precalculus Enhanced with Graphing Utilities

Question

In Problem, Write the partial fraction decomposition of each rational expression.

$\frac{6}{x(x-4)}$

Step-by-Step Solution

Answer

Partial fraction decomposition of a rational expression $\frac{6}{x(x-4)}$is

$\frac{6}{x(x-4)}=\frac{\frac{-3}{2}}{x}+\frac{\frac{3}{2}}{(x-4)}$

Step 1. Given data

The given rational expression is

$\frac{6}{x(x-4)}$

Step 2. Decomposition of rational expression

partial fraction decomposition of rational expression 

$$\begin{gathered} \frac{P\left(x\right)}{Q\left(x\right)}=\frac{A_1}{x-a_1}+\frac{A_2}{x-a_2}+\cdots +\frac{A_n}{x-a_n} \\ \frac{6}{x(x-4)}=\frac{A}{x}+\frac{B}{(x-4)} \\ \frac{6}{x(x-4)}=\frac{A(x-4)+Bx}{x(x-4)} \\ 6=A(x-4)+Bx \end{gathered}$$

Step 3. Value of numerators of partial fractions

Rearrange the equation

$$\begin{gathered} 6=A(x-4)+Bx \\ 0x+6=(A+B)x+(-4A) \end{gathered}$$

Equate the constants

$$\begin{gathered} -4A=6 \\ A=\frac{-3}{2} \end{gathered}$$

Equate the coefficients of x

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

So partial fraction decomposition is $\frac{6}{x(x-4)}=\frac{\frac{-3}{2}}{x}+\frac{\frac{3}{2}}{(x-4)}$

Step 4. Verification

Simplify the partial fractions

$$\begin{gathered} \frac{\frac{-3}{2}}{x}+\frac{\frac{3}{2}}{(x-4)}=\frac{\frac{-3}{2}(x-4)}{x(x-4)}+\frac{\frac{3}{2}x}{x(x-4)} \\ =\frac{-3x+12+3x}{2x(x-4)} \\ =\frac{12}{2x(x-4)} \\ =\frac{6}{x(x-4)} \end{gathered}$$

It is equal to the original expression so partial fraction decomposition is correct.