Chapter 11.5, Problem 26AYU

In problems 13-46, find the partial fraction decomposition of each rational expression. $\frac{x^2+x}{\left(x+2\right){\left(x-1\right)}^2}$

To determine

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

Answer to Problem 26AYU

Solution:

$\frac{2}{9\left(x + 2\right)} + \frac{7}{9\left(x - 1\right)} + \frac{2}{3{\left(x - 1\right)}^2}$

Explanation of Solution

Given: $\frac{x^2 + x}{\left(x + 2\right){\left(x - 1\right)}^2}$

Formula used:

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

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

where $A_1, A_2\dots A_n$ are to be determined.

b. If $Q$ has repeated linear factor of the form ${\left(x - a\right)}^n, \text{n} \ge 2$ an integer, then, in the partial fraction decomposition of $P\left(x\right)/Q\left(x\right)$, takes the form:

$\frac{P\left(x\right)}{Q\left(x\right)} = \frac{A_1}{\left(x - a\right)} + \frac{A_2}{{\left(x - a\right)}^2} + \cdots + \frac{A_n}{{\left(x - a\right)}^n}$

Where the numbers $A_1, A_2\dots A_n$ are to be determined.

Calculation:

The given partial fraction decomposition takes the form

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

$\begin{array}{l}\Rightarrow x^2 + x = A{\left(x - 1\right)}^2 + B\left(x - 1\right)\left(x + 2\right) + C\left(x + 2\right)\\ = A\left(x^2 - 2x + 1\right) + B\left(x^2 + x - 2\right) + C\left(x + 2\right)\end{array}$   -----Eq (1)

Equating the co-efficients on both sides in Eq(2),

$x^2, \Rightarrow A + B = 1$   -----Eq(4)

$x, \Rightarrow - 2A + B + C = 1$   -----Eq(5)

$\text{constant}\Rightarrow A - 2B + 2C = 0$   -----Eq(6)

Solving for $\text{A}, \text{B}, \text{C}$ we get

$\text{A} = \text{2/9}, \text{B} = \text{7/9}, \text{C} = \text{2/3}$

From Eq (1), the partial fraction decomposition of $\frac{x^2 + x}{\left(x + 2\right){\left(x - 1\right)}^2}$ is $\frac{2}{9\left(x + 2\right)} + \frac{7}{9\left(x - 1\right)} + \frac{2}{3{\left(x - 1\right)}^2}$