please help with b ### Partial Fraction Decomposition#### Objective: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.#### Problems and Solutions:**(a)** \(\frac{x - 12}{x^2 + x - 12}\)The partial fraction decomposition of this function is written as:\[ \frac{A}{(x - 3)} + \frac{B}{(x + 4)} \]This answer is marked with a green check mark, indicating that it is correct.**(b)** \(\frac{1}{x^2 + x^4}\)The incorrect form of the partial fraction decomposition is shown below:\[ \frac{A}{x^2 + x} + \frac{B}{x^2 + x^2} + \frac{C}{x^2 + x^3} + \frac{D}{x^2 + x^4} \]This answer is marked with a red cross, indicating that it is incorrect.**Explanation of Graphs and Diagrams:**- The diagram for problem (a) illustrates the correct partial fraction decomposition form. It consists of two terms, \(\frac{A}{(x - 3)}\) and \(\frac{B}{(x + 4)}\), merged by a plus sign.- The diagram for problem (b) displays an incorrect attempt at partial fraction decomposition with four terms, each with denominators \(x^2 + x\), \(x^2 + x^2\), \(x^2 + x^3\), and \(x^2 + x^4\), respectively. This form is not correct for the given function.**Note:**Partial fraction decomposition is a technique used to break down rational functions into simpler fractions. This is commonly utilized in integration and solving differential equations.

Convert partial fraction decomposition of the function

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. (a) (b) x 12 x² + x - 12 A B (x-3) (x+4) 1 A x² + x + + B + с D +². 4 X +x²

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. (a) (b) x 12 x² + x - 12 A B (x-3) (x+4) 1 A x² + x + + B + с D +². 4 X +x²

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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please help with b

### Partial Fraction Decomposition

#### Objective:
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.

#### Problems and Solutions:

**(a)** \(\frac{x - 12}{x^2 + x - 12}\)

The partial fraction decomposition of this function is written as:
\[ \frac{A}{(x - 3)} + \frac{B}{(x + 4)} \]

This answer is marked with a green check mark, indicating that it is correct.

**(b)** \(\frac{1}{x^2 + x^4}\)

The incorrect form of the partial fraction decomposition is shown below:
\[ \frac{A}{x^2 + x} + \frac{B}{x^2 + x^2} + \frac{C}{x^2 + x^3} + \frac{D}{x^2 + x^4} \]

This answer is marked with a red cross, indicating that it is incorrect.

**Explanation of Graphs and Diagrams:**

- The diagram for problem (a) illustrates the correct partial fraction decomposition form. It consists of two terms, \(\frac{A}{(x - 3)}\) and \(\frac{B}{(x + 4)}\), merged by a plus sign.

- The diagram for problem (b) displays an incorrect attempt at partial fraction decomposition with four terms, each with denominators \(x^2 + x\), \(x^2 + x^2\), \(x^2 + x^3\), and \(x^2 + x^4\), respectively. This form is not correct for the given function.

**Note:**
Partial fraction decomposition is a technique used to break down rational functions into simpler fractions. This is commonly utilized in integration and solving differential equations.

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