Answer
The partial fraction function is $\frac{3}{\left( x-3 \right)}-\frac{2}{\left( x-2 \right)}$.
Work Step by Step
Finding the partial fraction of the provided expression is given below,
$\frac{x}{\left( x-3 \right)\left( x-2 \right)}=\frac{A}{\left( x-3 \right)}+\frac{B}{\left( x-2 \right)}$
Then, multiply $\left( x-3 \right)\left( x-2 \right)$ on both sides,
$\begin{align}
& \left( x-3 \right)\left( x-2 \right)\frac{x}{\left( x-3 \right)\left( x-2 \right)}=\left( x-3 \right)\left( x-2 \right)\frac{A}{\left( x-3 \right)}+\left( x-3 \right)\left( x-2 \right)\frac{B}{\left( x-2 \right)} \\
& x=\left( x-2 \right)A+\left( x-3 \right)B \\
& =Ax-2A+Bx-3B \\
& x=x\left( A+B \right)-2A-3B
\end{align}$ …… (I)
Now, compare the coefficient of equation (I),
$A+B=1$ …… (II)
$-2A-3B=0$ …… (III)
And multiply equation (II) by 2 and add with equation (III),
Then
$\begin{align}
& -B=2 \\
& B=-2
\end{align}$
By putting the value of B in equation (II),
$\begin{align}
& A-2=1 \\
& A=3
\end{align}$
Therefore, the partial fraction is given below,
$\frac{x}{\left( x-3 \right)\left( x-2 \right)}=\frac{3}{\left( x-3 \right)}-\frac{2}{\left( x-2 \right)}$
Thus, the partial fraction of the given expression is, $\frac{3}{\left( x-3 \right)}-\frac{2}{\left( x-2 \right)}$.
