The Correct Form of Partial Fraction Decomposition

Oct 13, 2025 by windailys.com 84 views

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Introduction

Partial fraction decomposition is a powerful technique used in algebra to simplify complex rational expressions into more manageable parts. It involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate, differentiate, or solve. In this article, we will explore the correct form of the partial fraction decomposition for the expression $\frac{4x^3 + 3x^2}{(x+1)^2\left(x^2+7\right)^2}$ .

Understanding the Problem

The given expression is a rational function with a cubic polynomial in the numerator and a quadratic polynomial in the denominator, raised to the power of 2. To decompose this expression into partial fractions, we need to identify the factors of the denominator and create a corresponding set of partial fractions.

The Correct Form of Partial Fraction Decomposition

The correct form of the partial fraction decomposition for the given expression is:

$\frac{4x^3 + 3x^2}{(x+1)^2\left(x^2+7\right)^2} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{Cx+D}{x^2+7} + \frac{Ex+F}{\left(x^2+7\right)^2}$

This form consists of four partial fractions, each with a different denominator factor. The first two fractions have denominators of $(x+1)$ and $(x+1)^2$ , respectively, while the last two fractions have denominators of $x^2+7$ and $\left(x^2+7\right)^2$ , respectively.

Why This Form is Correct

To understand why this form is correct, let's consider the factors of the denominator. The denominator is a product of two quadratic polynomials, $(x+1)^2$ and $\left(x^2+7\right)^2$ . Since the first polynomial is raised to the power of 2, we need to create a corresponding partial fraction with a denominator of $(x+1)^2$ . Similarly, since the second polynomial is also raised to the power of 2, we need to create another partial fraction with a denominator of $\left(x^2+7\right)^2$ .

Comparison with Other Options

Let's compare the correct form with the other options:

Option A: $\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{Cx+D}{x^2+7} + \frac{Ex+F}{\left(x^2+7\right)^2}$
Option B: $\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{Cx+D}{x^2+7} + \frac{E}{\left(x^2+7\right)^2}$
Option C: $\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{Cx}{x^2+7} + \frac{D}{\left(x^2+7\right)^2}$

Option A is the correct form, as it includes all the necessary partial fractions with the correct denominators. Option B is incorrect because it only includes a single partial fraction with a denominator of $\left(x^2+7\right)^2$ , whereas we need two partial fractions with denominators of $x^2+7$ and $\left(x^2+7\right)^2$ . Option C is also incorrect because it only includes a single partial fraction with a denominator of $x^2+7$ , whereas we need two partial fractions with denominators of $x^2+7$ and $\left(x^2+7\right)^2$ .

Conclusion

In conclusion, the correct form of the partial fraction decomposition for the expression $\frac{4x^3 + 3x^2}{(x+1)^2\left(x^2+7\right)^2}$ is:

$\frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{Cx+D}{x^2+7} + \frac{Ex+F}{\left(x^2+7\right)^2}$

Example Use Case

Partial fraction decomposition is a powerful technique used in algebra to simplify complex rational expressions into more manageable parts. It involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate, differentiate, or solve. For example, consider the expression $\frac{x^2+2x+1}{(x+1)^2(x+2)}$ . We can decompose this expression into partial fractions as follows:

$\frac{x^2+2x+1}{(x+1)^2(x+2)} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x+2}$

Solving for the unknown coefficients $A$ , $B$ , and $C$ , we get:

$A = 1$ $B = -1$ $C = 1$

Therefore, the partial fraction decomposition of the expression is:

$\frac{x^2+2x+1}{(x+1)^2(x+2)} = \frac{1}{x+1} - \frac{1}{(x+1)^2} + \frac{1}{x+2}$

This decomposition can be used to integrate, differentiate, or solve the expression more easily.

Tips and Tricks

Here are some tips and tricks to help you with partial fraction decomposition:

Make sure to identify all the factors of the denominator and create a corresponding set of partial fractions.
Use the correct form of partial fraction decomposition, which includes all the necessary partial fractions with the correct denominators.
Solve for the unknown coefficients by equating the numerators and denominators of the original expression and the partial fraction decomposition.
Use algebraic manipulations to simplify the expression and make it easier to work with.

By following these tips and tricks, you can master the art of partial fraction decomposition and simplify complex rational expressions with ease.

Common Mistakes

Here are some common mistakes to avoid when working with partial fraction decomposition:

Failing to identify all the factors of the denominator and creating an incomplete set of partial fractions.
Using the incorrect form of partial fraction decomposition, which can lead to incorrect solutions.
Failing to solve for the unknown coefficients, which can lead to incorrect solutions.
Not using algebraic manipulations to simplify the expression, which can make it harder to work with.

By avoiding these common mistakes, you can ensure that your partial fraction decomposition is correct and accurate.

Conclusion

In conclusion, partial fraction decomposition is a powerful technique used in algebra to simplify complex rational expressions into more manageable parts. It involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate, differentiate, or solve. By following the correct form of partial fraction decomposition and using algebraic manipulations, you can master the art of partial fraction decomposition and simplify complex rational expressions with ease.

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Introduction

Partial fraction decomposition is a powerful technique used in algebra to simplify complex rational expressions into more manageable parts. It involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate, differentiate, or solve. In this article, we will answer some frequently asked questions about partial fraction decomposition.

Q: What is partial fraction decomposition?

A: Partial fraction decomposition is a technique used to simplify complex rational expressions into more manageable parts. It involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate, differentiate, or solve.

Q: When should I use partial fraction decomposition?

A: You should use partial fraction decomposition when you have a complex rational expression that you want to simplify. This can be useful when you need to integrate, differentiate, or solve the expression.

Q: How do I know if I need to use partial fraction decomposition?

A: You need to use partial fraction decomposition if the denominator of the rational expression has multiple factors or if the numerator has a degree greater than the denominator.

Q: What are the steps to perform partial fraction decomposition?

A: The steps to perform partial fraction decomposition are:

Identify the factors of the denominator.
Create a corresponding set of partial fractions.
Solve for the unknown coefficients.
Simplify the expression.

Q: How do I identify the factors of the denominator?

A: To identify the factors of the denominator, you need to factor the denominator into its prime factors. This can be done using various techniques such as factoring by grouping, factoring by difference of squares, or factoring by sum and difference of cubes.

Q: What is the correct form of partial fraction decomposition?

A: The correct form of partial fraction decomposition is:

$\frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c} + \frac{D}{x-d}$

where $a$ , $b$ , $c$ , and $d$ are the factors of the denominator.

Q: How do I solve for the unknown coefficients?

A: To solve for the unknown coefficients, you need to equate the numerators and denominators of the original expression and the partial fraction decomposition. This can be done using algebraic manipulations such as cross-multiplication or substitution.

Q: What are some common mistakes to avoid when performing partial fraction decomposition?

A: Some common mistakes to avoid when performing partial fraction decomposition include:

Failing to identify all the factors of the denominator.
Using the incorrect form of partial fraction decomposition.
Failing to solve for the unknown coefficients.
Not using algebraic manipulations to simplify the expression.

Q: How do I know if my partial fraction decomposition is correct?

A: To know if your partial fraction decomposition is correct, you need to check that the original expression and the partial fraction decomposition are equivalent. This can be done by equating the numerators and denominators of the two expressions and simplifying.

Q: Can I use partial fraction decomposition to solve systems of equations?

A: Yes, you can use partial fraction decomposition to solve systems of equations. This can be done by expressing each equation as a rational function and then using partial fraction decomposition to simplify the expression.

Q: Are there any limitations to partial fraction decomposition?

A: Yes, there are some limitations to partial fraction decomposition. For example, it may not be possible to decompose a rational function into partial fractions if the denominator has repeated factors or if the numerator has a degree greater than the denominator.

Conclusion

In conclusion, partial fraction decomposition is a powerful technique used in algebra to simplify complex rational expressions into more manageable parts. It involves expressing a rational function as a sum of simpler fractions, which can be easier to integrate, differentiate, or solve. By following the correct steps and avoiding common mistakes, you can master the art of partial fraction decomposition and simplify complex rational expressions with ease.

Example Use Case

Partial fraction decomposition can be used to solve systems of equations. For example, consider the following system of equations:

$\begin{cases} \frac{x^2+2x+1}{(x+1)^2(x+2)} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x+2} \\ \frac{x^2+3x+2}{(x+1)^2(x+2)} = \frac{D}{x+1} + \frac{E}{(x+1)^2} + \frac{F}{x+2} \end{cases}$

We can use partial fraction decomposition to solve for the unknown coefficients $A$ , $B$ , $C$ , $D$ , $E$ , and $F$ . By equating the numerators and denominators of the two equations and simplifying, we can find the values of the unknown coefficients.

Tips and Tricks

Here are some tips and tricks to help you with partial fraction decomposition:

Make sure to identify all the factors of the denominator and create a corresponding set of partial fractions.
Use the correct form of partial fraction decomposition, which includes all the necessary partial fractions with the correct denominators.
Solve for the unknown coefficients by equating the numerators and denominators of the original expression and the partial fraction decomposition.
Use algebraic manipulations to simplify the expression and make it easier to work with.

By following these tips and tricks, you can master the art of partial fraction decomposition and simplify complex rational expressions with ease.

Common Mistakes

Here are some common mistakes to avoid when working with partial fraction decomposition:

Failing to identify all the factors of the denominator and creating an incomplete set of partial fractions.
Using the incorrect form of partial fraction decomposition, which can lead to incorrect solutions.
Failing to solve for the unknown coefficients, which can lead to incorrect solutions.
Not using algebraic manipulations to simplify the expression, which can make it harder to work with.

By avoiding these common mistakes, you can ensure that your partial fraction decomposition is correct and accurate.