Q. 17 - Precalculus Enhanced with Graphing Utilities

Question

Write the partial fraction decomposition of each rational expression:

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

Step-by-Step Solution

Answer

The partial fraction decomposition is,

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

Step 1. Given Information

The rational expression is,

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

Step 2. Decompose the partial fractions.

Decompose the rational expressions,

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

Multiplying both sides by,

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

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

$$\begin{gathered} A+B=1 \\ -2A-B=0 \end{gathered}$$

Solving the equations we get,

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

The partial fraction decomposition is,

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

Step 3. Checking the solution.

Adding the rational expressions,

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

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