Answer
The partial fraction decomposition contains a term of the form:
$\frac{Ax+B}{a{{x}^{2}}+bx+c}$, where $a{{x}^{2}}+bx+c\ne 0$ when the denominator of a rational function is with a prime quadratic factor of the form of $\left( a{{x}^{2}}+bx+c \right)$.
Work Step by Step
We know the denominator is in the form of a prime quadratic factor.
Consider an example,
$\frac{3x}{x\left( {{x}^{2}}+5 \right)}$
And the partial fraction decomposition of the rational function is:
$\frac{3x}{x\left( {{x}^{2}}+5 \right)}=\frac{A}{x}+\frac{B}{\left( {{x}^{2}}+5 \right)}$
And by multiplying both sides by $x\left( {{x}^{2}}+5 \right)$:
$\begin{align}
& 3x=A\left( {{x}^{2}}+5 \right)+Bx \\
& 3x=A{{x}^{2}}+5A+Bx
\end{align}$
Equating the like terms of the equation:
$\begin{align}
& A=0 \\
& B=3
\end{align}$
Put these values in $\frac{3x}{x\left( {{x}^{2}}+5 \right)}=\frac{A}{x}+\frac{B}{\left( {{x}^{2}}+5 \right)}$.
Thus, $\frac{3x}{x\left( {{x}^{2}}+5 \right)}=\frac{3}{\left( {{x}^{2}}+5 \right)}$.
