Chapter B, Problem 1P

To determine

The partial fraction decomposition of the given rational function.

Answer to Problem 1P

Solution:

The partial fraction decomposition of the given rational function is $\frac{5}{x+2}-\frac{3}{x+1}$.

Explanation of Solution

Given:

The given rational function is,

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

Approach:

The standard way to find the partial fraction of the rational function,

1. Determine the general form of the partial fraction decomposition of $p\left(x\right)/q\left(x\right)$.

2. Multiply both the resulting decomposition by $q\left(x\right)$.

3. Equate the equation in order to determine the constants in the partial fraction decomposition.

Calculation:

Consider the given rational function,

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

Write the general form of the partial fraction decomposition of the above given rational function.

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

Multiply both sides of this equation by $\left(x+1\right)\left(x+2\right)$ yields.

$2x-1=A\left(x+2\right)+B\left(x+1\right)$

Equate the coefficient of like powers of $x$ on both sides of the above equation to obtain the $A$ and $B$.

$A+B=2$ and $2A+B=-1$.

Consequently,

$A=-3\text{and}B=5$

Substitute the values of $A$ and $B$ in the general form of the partial fraction.

$\begin{array}{rll}\frac{2x-1}{\left(x+1\right)\left(x+2\right)}& =& -\frac{3}{x+1}+\frac{5}{x+2}\\ & =& \frac{5}{x+2}-\frac{3}{x+1}\end{array}$

Therefore, the partial fraction decomposition of the given rational function is $\frac{5}{x+2}-\frac{3}{x+1}$.

Conclusion:

Hence, the partial fraction decomposition of the given rational function is $\frac{5}{x+2}-\frac{3}{x+1}$.