$\"\"$

\

The improper rational expression is $\"\"$ .

\

The degree of numerator and the denominator of the expression are equal, hence first the numerator with the denominator.

\

$\"\"$

\

The expression can be written as $\"\"$ .

\

Rewrite the expression into partial fraction.

\

$\"\"$

\

Find the values of $\"\"$ and $\"\"$ .

\

Multiple each side by the denominator fraction.

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

Equate the coefficent of $\"\"$ and constant.

\

$\"\"$ and $\"\"$ .

\

$\"\"$

\

System of equations are $\"\"$ and $\"\"$ .

\

Write the equations into matrix form $\"\"$ , where $\"\"$ is coefficient matrix, $\"\"$ is variable matrix and $\"\"$ is constant matrix.

\

\

$\"\"$ .

\

Find variable matrix $\"\"$ .

\

$\"\"$ .

\

Inverse of a $\"\"$ matrix : $\"\"$ .

\

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ and $\"\"$ .

\

Substistute the $\"\"$ and $\"\"$ values in equation $\"\"$ .

\

$\"\"$

\

The partial decomposed function is $\"\"$ .

\

$\"\"$

\

The partial decomposed function is $\"\"$ .