Question

Write the partial fraction decomposition of each rational expression:

$\frac{4}{x(x-1)}$.

Short answer

The partial fraction decomposition is,

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

Step-by-step solution

Step 1. Given Information.

The rational expression is,

$\frac{4}{x(x-1)}$.

Step 2. Partial Decomposition.

Decompose the rational expression,

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

Multiplying each side by $x(x-1)$,

role="math" localid="1647369079514" $$\begin{gathered} 4=A(x-1)+Bx \\ 4=(A+B)x-A..........\left(2\right) \end{gathered}$$

Equating the coefficients of like powers, we get,

$$\begin{gathered} A+B=0 \\ -A=4 \end{gathered}$$

Solving the two equations,

$$\begin{gathered} A=-4 \\ B=4 \end{gathered}$$

The partial fraction decomposition is,

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

Step 3. Checking the solution.

Adding the rational expressions,

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

$$\begin{gathered} =\frac{-4x+4+4x}{x(x-1)} \\ =\frac{4}{x(x-1)} \end{gathered}$$