91Ó°ÊÓ

Any number which can be easily represented in the form of $\mathrm{p}/\mathrm{q}$, such that localid="1662726245663" $\mathrm{p}$and localid="1662726249923" $\mathrm{q}$are integers and localid="1662726253823" role="math" $\mathrm{q}≠0$ is known as a rational number.

Similarly, we can define a rational function as the ratio of two polynomial functions $\mathrm{P}\left(\mathrm{x}\right)$and $\mathrm{Q}\left(\mathrm{x}\right)$, where localid="1662726263552" $\mathrm{P}$and localid="1662726260804" $\mathrm{Q}$are polynomials in localid="1662726257626" $\mathrm{x}$ and localid="1662726267995" $\mathrm{Q}\left(\mathrm{x}\right)≠0$.

A rational function is known as proper if the degree of localid="1662726271465" $\mathrm{P}\left(\mathrm{x}\right)$is less than the degree of $\mathrm{Q}\left(\mathrm{x}\right)$; otherwise, it is known as an improper rational function.

With the help of the long division process, we can reduce improper rational functions to proper rational functions. Therefore, if $\text{P}\left(\text{x}\right)/\text{Q}\left(\text{x}\right)$is improper, then it can be expressed as:

$\frac{\mathrm{P}\left(\mathrm{x}\right)}{\mathrm{Q}\left(\mathrm{x}\right)}=\mathrm{A}\left(\mathrm{x}\right)+\frac{\mathrm{R}\left(\mathrm{x}\right)}{\mathrm{Q}\left(\mathrm{x}\right)}$

Here,localid="1662726284688" $\mathrm{A}\left(\mathrm{x}\right)$is a polynomial in localid="1662726275840" $\mathrm{x}$and localid="1662726279992" $\mathrm{R}\left(\mathrm{x}\right)/\mathrm{Q}\left(\mathrm{x}\right)$is a proper rational function.

The given rational function is $\frac{−8{\mathrm{s}}^2−5\mathrm{s}+9}{(\mathrm{s}+1)({\mathrm{s}}^2−3\mathrm{s}+2)}$

Rewrite $\frac{−8{\mathrm{s}}^2−5\mathrm{s}+9}{(\mathrm{s}+1)({\mathrm{s}}^2−3\mathrm{s}+2)}$as a sum of partial fractions as:

$\frac{−8{\mathrm{s}}^2−5\mathrm{s}+9}{(\mathrm{s}+1)(\mathrm{s}−2)(\mathrm{s}−1)}=\frac{\mathrm{A}}{\mathrm{s}+1}+\frac{\mathrm{B}}{\mathrm{s}−2}+\frac{\mathrm{C}}{\mathrm{s}−1}$

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

$−8{\mathrm{s}}^2−5\mathrm{s}+9=\mathrm{A}(\mathrm{s}−2)(\mathrm{s}−1)+\mathrm{B}(\mathrm{s}+1)(\mathrm{s}−1)+\mathrm{C}(\mathrm{s}+1)(\mathrm{s}−2)$

Find the constants as:

For $\mathrm{s}=−1,−8{(−1)}^2−5(−1)+9=\mathrm{A}(−3)(−2)⇒\mathrm{A}=1$

For $\mathrm{s}=2:−8{\left(2\right)}^2−5\left(2\right)+9=\mathrm{B}\left(3\right)\left(1\right)⇒\mathrm{B}=−11$.

For $\mathrm{s}=1:−8{\left(1\right)}^2−5\left(1\right)+9=\mathrm{C}\left(2\right)(−1)⇒\mathrm{C}=2$.

Substitute the value of constants into$\frac{−8{\mathrm{s}}^2−5\mathrm{s}+9}{(\mathrm{s}+1)(\mathrm{s}−2)(\mathrm{s}−1)}=\frac{\mathrm{A}}{\mathrm{s}+1}+\frac{\mathrm{B}}{\mathrm{s}−2}+\frac{\mathrm{C}}{\mathrm{s}−1}$ as follows:

$\begin{array}{c}\frac{−8{\mathrm{s}}^2−5\mathrm{s}+9}{(\mathrm{s}+1)(\mathrm{s}−2)(\mathrm{s}−1)}=\frac{1}{\mathrm{s}+1}+\frac{−11}{\mathrm{s}−2}+\frac{2}{\mathrm{s}−1}\\ =\frac{1}{\mathrm{s}+1}−\frac{11}{\mathrm{s}−2}+\frac{2}{\mathrm{s}−1}\end{array}$

Therefore, the partial fraction expansion for the given rational function is $\frac{1}{\mathrm{s}+1}−\frac{11}{\mathrm{s}−2}+\frac{2}{\mathrm{s}−1}$.