Answer
The partial fraction is $\frac{7}{\left( x-9 \right)}-\frac{4}{\left( x+2 \right)}$.
Work Step by Step
We find the partial fraction of the provided expression as given below,
$\frac{3x+50}{\left( x-9 \right)\left( x+2 \right)}=\frac{A}{\left( x-9 \right)}+\frac{B}{\left( x+2 \right)}$
Then, multiply $\left( x-9 \right)\left( x+2 \right)$ on both sides,
$\begin{align}
& \left( x-9 \right)\left( x+2 \right)\frac{3x+50}{\left( x-9 \right)\left( x+2 \right)}=\left( x-9 \right)\left( x+2 \right)\frac{A}{\left( x-9 \right)}+\left( x-9 \right)\left( x+2 \right)\frac{B}{\left( x+2 \right)} \\
& 3x+50=\left( x+2 \right)A+\left( x-9 \right)B \\
& =Ax+2A+Bx-9B \\
& 3x+50=x\left( A+B \right)+2A-9B
\end{align}$ …… (I)
Now, compare the coefficient of equation (I),
$A+B=3$ …… (II)
$2A-9B=50$ …… (III)
Then, multiply equation (II) by 2 and subtract from equation (III),
Then,
$11B=44$
$\begin{align}
& B=\frac{44}{-11} \\
& =-4
\end{align}$
Now, put the value in equation (II),
$\begin{align}
& A-4=3 \\
& =7
\end{align}$
Therefore,
$\frac{3x+50}{\left( x-9 \right)\left( x+2 \right)}=\frac{7}{\left( x-9 \right)}-\frac{4}{\left( x+2 \right)}$
Thus, the partial fraction of the given expression is $\frac{7}{\left( x-9 \right)}-\frac{4}{\left( x+2 \right)}$.
