91Ó°ÊÓ

• The partial fraction expansion of a rational fraction (that is, a fraction with both a polynomial numerator and a polynomial denominator) is an algebraic procedure that involves expressing the fraction as a sum of a polynomial plus one or more fractions with a simpler denominator.
• The partial fraction decomposition is important because it gives methods for a variety of rational function computations, such as explicit antiderivative computations, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms.

The given rational function is $\frac{−2{\mathrm{s}}^2−3\mathrm{s}−2}{\mathrm{s}{(\mathrm{s}+1)}^2}$.

Rewrite $\frac{−2{\mathrm{s}}^2−3\mathrm{s}−2}{\mathrm{s}{(\mathrm{s}+1)}^2}$ as a sum of partial fractions as:

$\frac{−2{\mathrm{s}}^2−3\mathrm{s}−2}{\mathrm{s}{(\mathrm{s}+1)}^2}=\frac{\mathrm{A}}{\mathrm{s}}+\frac{\mathrm{B}}{(\mathrm{s}+1)}+\frac{\mathrm{C}}{{(\mathrm{s}+1)}^2}$

Multiply both sides by the LCD $(\mathrm{s}+1)(\mathrm{s}−2)$as follows:

$\begin{array}{c}−2{\mathrm{s}}^2−3\mathrm{s}−2=\mathrm{A}{\left(\mathrm{s}+1\right)}^2+\mathrm{Bs}(\mathrm{s}+1)+\mathrm{Cs}\\ −2{\mathrm{s}}^2−3\mathrm{s}−2={\mathrm{As}}^2+2\mathrm{As}+\mathrm{A}+{\mathrm{Bs}}^2+\mathrm{Bs}+\mathrm{Cs}\end{array}$

Group like terms and factor as:

Compare coefficients of ${\mathrm{s}}^2$, $\mathrm{s}$ and constant term as follows:

$\begin{array}{c}\mathrm{A}+\mathrm{B}=−2\\ 2\mathrm{A}+\mathrm{B}+\mathrm{C}=−3\\ \mathrm{A}=−2\end{array}$

Find the constants as:

$\mathrm{A}=−2,\mathrm{B}=0,\mathrm{C}=1$

Substitute the value of constants into $\frac{−2{\mathrm{s}}^2−3\mathrm{s}−2}{\mathrm{s}{(\mathrm{s}+1)}^2}=\frac{\mathrm{A}}{\mathrm{s}}+\frac{\mathrm{B}}{(\mathrm{s}+1)}+\frac{\mathrm{C}}{{(\mathrm{s}+1)}^2}$ as:

$\begin{array}{l}\frac{−2{\mathrm{s}}^2−3\mathrm{s}−2}{\mathrm{s}{(\mathrm{s}+1)}^2}=\frac{−2}{\mathrm{s}}+\frac{0}{(\mathrm{s}+1)}+\frac{1}{{(\mathrm{s}+1)}^2}\\ \frac{−2{\mathrm{s}}^2−3\mathrm{s}−2}{\mathrm{s}{(\mathrm{s}+1)}^2}=\frac{1}{{(\mathrm{s}+1)}^2}−\frac{2}{\mathrm{s}}\end{array}$

Therefore, the partial fraction expansion for the given ration function is $\frac{1}{{(\mathrm{s}+1)}^2}−\frac{2}{\mathrm{s}}$.