Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

In Problems 11–20, determine the partial fraction expansion for the given rational function.
$\frac{- 8 s^{2} - 5 s + 9}{(s + 1) (s^{2} - 3 s + 2)}$

The partial fraction expansions for the given rational function $\frac{- 8 s^{2} - 5 s + 9}{(s + 1) (s^{2} - 3 s + 2)}$ is $\frac{1}{s + 1} - \frac{11}{s - 2} + \frac{2}{s - 1}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of partial fraction expansion

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

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

A rational function is known as proper if the degree of localid="1662726271465" $P (x)$ is less than the degree of $Q (x)$ ; 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 $P (x) / Q (x)$ is improper, then it can be expressed as:

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

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

Step 2: Determine the partial fraction expansion for the given rational function

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

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

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

Multiply both sides by the LCD $(s + 1) (s - 2) (s - 1)$ as follows:

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

Find the constants as:

For $s = - 1, - 8 {(- 1)}^{2} - 5 (- 1) + 9 = A (- 3) (- 2) \Rightarrow A = 1$

For $s = 2 : - 8 {(2)}^{2} - 5 (2) + 9 = B (3) (1) \Rightarrow B = - 11$ .

For $s = 1 : - 8 {(1)}^{2} - 5 (1) + 9 = C (2) (- 1) \Rightarrow C = 2$ .

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

$\begin{matrix} \frac{- 8 s^{2} - 5 s + 9}{(s + 1) (s - 2) (s - 1)} = \frac{1}{s + 1} + \frac{- 11}{s - 2} + \frac{2}{s - 1} \\ = \frac{1}{s + 1} - \frac{11}{s - 2} + \frac{2}{s - 1} \end{matrix}$

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

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

In Problems 11–20, determine the partial fraction expansion for the given rational function.
$\frac{- 8 s^{2} - 5 s + 9}{(s + 1) (s^{2} - 3 s + 2)}$

Step by Step Solution

Step 1: Definition of partial fraction expansion

Step 2: Determine the partial fraction expansion for the given rational function

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 11–20, determine the partial fraction expansion for the given rational function.−8s2−5s+9(s+1)(s2−3s+2)

Step by Step Solution

Step 1: Definition of partial fraction expansion

Step 2: Determine the partial fraction expansion for the given rational function

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In Problems 11–20, determine the partial fraction expansion for the given rational function.
$\frac{- 8 s^{2} - 5 s + 9}{(s + 1) (s^{2} - 3 s + 2)}$