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 $\frac{{\mathrm{s}}^2−26\mathrm{s}−47}{(\mathrm{s}−1)(\mathrm{s}+2)(\mathrm{s}+5)}$

Rewrite $\frac{{\mathrm{s}}^2−26\mathrm{s}−47}{(\mathrm{s}−1)(\mathrm{s}+2)(\mathrm{s}+5)}$as a sum of partial fractions as:

$\frac{{\mathrm{s}}^2−26\mathrm{s}−47}{(\mathrm{s}−1)(\mathrm{s}+2)(\mathrm{s}+5)}=\frac{\mathrm{A}}{\mathrm{s}−1}+\frac{\mathrm{B}}{\mathrm{s}+2}+\frac{\mathrm{C}}{\mathrm{s}+5}$

Multiply both sides by the LCD$(\mathrm{s}−1)(\mathrm{s}+2)(\mathrm{s}+5)$as follows:

${\mathrm{s}}^2−26\mathrm{s}−47=\mathrm{A}(\mathrm{s}+2)(\mathrm{s}+5)+\mathrm{B}(\mathrm{s}−1)(\mathrm{s}+5)+\mathrm{C}(\mathrm{s}−1)(\mathrm{s}+2)$

Find the constants as:

For$\mathrm{s}=1,{\left(1\right)}^2−26\left(1\right)−47=\mathrm{A}\left(3\right)\left(6\right)⇒\mathrm{A}=−4$.

For$\mathrm{s}=−2,{(−2)}^2−26(−2)−47=\mathrm{B}(−3)\left(3\right)⇒\mathrm{B}=−1$.

For $\mathrm{s}=−5,{(−5)}^2−26(−5)−47=\mathrm{C}(−6)(−3)⇒\mathrm{C}=6$.

Substitute the value of constants into $\frac{{\mathrm{s}}^2−26\mathrm{s}−47}{(\mathrm{s}−1)(\mathrm{s}+2)(\mathrm{s}+5)}=\frac{\mathrm{A}}{\mathrm{s}−1}+\frac{\mathrm{B}}{\mathrm{s}+2}+\frac{\mathrm{C}}{\mathrm{s}+5}$ as follows:

$\begin{array}{c}\frac{{\mathrm{s}}^2−26\mathrm{s}−47}{(\mathrm{s}−1)(\mathrm{s}+2)(\mathrm{s}+5)}=\frac{−4}{\mathrm{s}−1}+\frac{−1}{\mathrm{s}+2}+\frac{6}{\mathrm{s}+5}\\ =\frac{6}{\mathrm{s}+5}−\frac{1}{\mathrm{s}+2}−\frac{4}{\mathrm{s}−1}\end{array}$

Therefore, the partial fraction expansion for the given rational function is

$\frac{6}{\mathrm{s}+5}−\frac{1}{\mathrm{s}+2}−\frac{4}{\mathrm{s}−1}$