Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficient (a) (b) x6 x² - 4 A x 2 + B x + 2 x4 (x²-x + 1)(x² + 8)² Ax + B x²-x-1 X + Cx + D x² +8 + Ex + F (x²+8)² 12 X
Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficient (a) (b) x6 x² - 4 A x 2 + B x + 2 x4 (x²-x + 1)(x² + 8)² Ax + B x²-x-1 X + Cx + D x² +8 + Ex + F (x²+8)² 12 X
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Example
**Problem Statement:**
Write out the form of the partial fraction decomposition of the function
\[
\frac{x^3 + x^2 + 1}{x(x - 1)(x^2 + x + 1)(x^2 + 1)^3}
\]
### Solution
We need to decompose the given rational function into partial fractions:
\[
\frac{x^3 + x^2 + 1}{x(x - 1)(x^2 + x + 1)(x^2 + 1)^3}
\]
The partial fraction decomposition can be expressed as:
\[
\frac{A}{x} + \frac{B}{x - 1} + \frac{Cx + D}{x^2 + x + 1} + \frac{Ex + F}{x^2 + 1} + \frac{Gx + H}{(x^2 + 1)^2} + \frac{Ix + J}{(x^2 + 1)^3}
\]
Here, \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\), \(I\), and \(J\) are constants to be determined.
This form sets up the structure required to solve the partial fraction decomposition systematically by equating coefficients and solving for these constants.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4971f23-5570-4fed-8cd6-5e1a07d18a42%2F96e45240-6a94-4d74-9c39-de6307e53d06%2Fae59ih7_processed.png&w=3840&q=75)
Transcribed Image Text:### Example
**Problem Statement:**
Write out the form of the partial fraction decomposition of the function
\[
\frac{x^3 + x^2 + 1}{x(x - 1)(x^2 + x + 1)(x^2 + 1)^3}
\]
### Solution
We need to decompose the given rational function into partial fractions:
\[
\frac{x^3 + x^2 + 1}{x(x - 1)(x^2 + x + 1)(x^2 + 1)^3}
\]
The partial fraction decomposition can be expressed as:
\[
\frac{A}{x} + \frac{B}{x - 1} + \frac{Cx + D}{x^2 + x + 1} + \frac{Ex + F}{x^2 + 1} + \frac{Gx + H}{(x^2 + 1)^2} + \frac{Ix + J}{(x^2 + 1)^3}
\]
Here, \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\), \(I\), and \(J\) are constants to be determined.
This form sets up the structure required to solve the partial fraction decomposition systematically by equating coefficients and solving for these constants.
). Do not determine the numerical values of the coefficients.
### Problem (a)
\[ \frac{x^6}{x^2 - 4} \]
**Incorrect Decomposition:**
\[ \frac{A}{x - 2} + \frac{B}{x + 2} \]
Explanation: This decomposition is marked incorrect. The correct partial fraction decomposition needs to be implemented according to the polynomial degree and the factors of the denominator.
### Problem (b)
\[ \frac{x^4}{(x^2 - x + 1)(x^2 + 8)^2} \]
**Incorrect Decomposition:**
\[ \frac{Ax + B}{x^2 - x - 1} + \frac{Cx + D}{x^2 + 8} + \frac{Ex + F}{(x^2 + 8)^2} \]
Explanation: This decomposition is marked incorrect. The correct partial fraction decomposition needs to use appropriate forms taking into account the factorization and the exponents in the denominator.
**Note for Students:** This image includes examples where partial fraction decomposition is done incorrectly. When doing partial fraction decomposition, always ensure that the form of the decomposition matches the given function's denominator factorization and polynomial degrees.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4971f23-5570-4fed-8cd6-5e1a07d18a42%2F96e45240-6a94-4d74-9c39-de6307e53d06%2F618s99q_processed.png&w=3840&q=75)
Transcribed Image Text:**Partial Fraction Decomposition**
**Instructions:** Write out the form of the partial fraction decomposition of the function (as in [this example](#)). Do not determine the numerical values of the coefficients.
### Problem (a)
\[ \frac{x^6}{x^2 - 4} \]
**Incorrect Decomposition:**
\[ \frac{A}{x - 2} + \frac{B}{x + 2} \]
Explanation: This decomposition is marked incorrect. The correct partial fraction decomposition needs to be implemented according to the polynomial degree and the factors of the denominator.
### Problem (b)
\[ \frac{x^4}{(x^2 - x + 1)(x^2 + 8)^2} \]
**Incorrect Decomposition:**
\[ \frac{Ax + B}{x^2 - x - 1} + \frac{Cx + D}{x^2 + 8} + \frac{Ex + F}{(x^2 + 8)^2} \]
Explanation: This decomposition is marked incorrect. The correct partial fraction decomposition needs to use appropriate forms taking into account the factorization and the exponents in the denominator.
**Note for Students:** This image includes examples where partial fraction decomposition is done incorrectly. When doing partial fraction decomposition, always ensure that the form of the decomposition matches the given function's denominator factorization and polynomial degrees.
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