Chapter 11.5, Problem 13AYU

In problems 13-46, find the partial fraction decomposition of each rational expression. $\frac{4}{x\left(x-1\right)}$

To determine

To find: The partial fraction decomposition of the given rational function.

Answer to Problem 13AYU

Solution:

$\frac{ - 4}{x} + \frac{4}{x - 1}$

Explanation of Solution

Given:

$\frac{4}{x\left(x - 1\right)}$

Formula used:

If $Q(x) = (x - a_1)\mathrm{...}\mathrm{...}(x - a_n)$ then partial fraction decomposition of $\text{P}(\text{x})/\text{Q}(\text{x})$ takes the form:

$\frac{P\left(x\right)}{Q\left(x\right)} = \frac{A_1}{\left(x - a_1\right)} + \frac{A_2}{\left(x - a_2\right)} + \dots \frac{A_n}{\left(x - a_n\right)}$

Where $A_1, A_2\mathrm{...}{A}_n$ are to be determined.

Calculation:

$\begin{array}{rrr}\hfill \frac{4}{x\left(x - 1\right)}& = \hfill & \frac{A}{x} + \frac{B}{\left(x - 1\right)}\hfill \\ \hfill 4& \hfill = & A\left(x - 1\right) + Bx\hfill \\ \hfill & = \hfill & Ax - A + Bx\hfill \end{array}$…. Eq (1)

Equating the coefficients of $x$ and constant we get

$A + B = 0 \text{and} - A = 4$

Hence $A = - 4 \text{and} B = 4;$

From Eq (1), the partial fraction decomposition of $\frac{4}{x\left(x - 1\right)}$ is $\frac{ - 4}{x} + \frac{4}{x - 1}$.