For The Expression $\frac{15 X^3+25 X^2+5 X+2}{\left(5 X^2+1\right)^2}$, Which Sum Represents The Correct Form Of The Partial Fraction Decomposition?A. $\frac{A}{5 X^2+1}+\frac{B}{\left(5 X^2+1\right)^2}$B. $\frac{A X+B}{5

Introduction

Partial fraction decomposition is a powerful technique used in algebra to break down complex rational functions into simpler fractions. This method is essential in solving equations, simplifying expressions, and analyzing functions. In this article, we will explore the concept of partial fraction decomposition and apply it to the given expression $\frac{15 x^3+25 x^2+5 x+2}{\left(5 x^2+1\right)^2}$.

What is Partial Fraction Decomposition?

Partial fraction decomposition is a process of expressing a rational function as a sum of simpler fractions. The rational function is typically written in the form of a quotient of two polynomials, where the numerator is a polynomial of lower degree than the denominator. The goal is to express the rational function as a sum of fractions, where each fraction has a polynomial numerator and a polynomial denominator.

The General Form of Partial Fraction Decomposition

The general form of partial fraction decomposition is:

$$\frac{P(x)}{Q(x)} = \frac{A_1}{Q_1(x)} + \frac{A_2}{Q_2(x)} + \cdots + \frac{A_n}{Q_n(x)}$$

where $P(x)$ is the numerator, $Q(x)$ is the denominator, and $A_1, A_2, \ldots, A_n$ are constants to be determined.

Applying Partial Fraction Decomposition to the Given Expression

To apply partial fraction decomposition to the given expression $\frac{15 x^3+25 x^2+5 x+2}{\left(5 x^2+1\right)^2}$, we need to follow these steps:

1. Factor the denominator: The denominator is already factored as $\left(5 x^2+1\right)^2$.
2. Write the partial fraction decomposition: We can write the partial fraction decomposition as:

$$\frac{15 x^3+25 x^2+5 x+2}{\left(5 x^2+1\right)^2} = \frac{A}{5 x^2+1} + \frac{B}{\left(5 x^2+1\right)^2}$$

where $A$ and $B$ are constants to be determined.

Solving for A and B

To solve for $A$ and $B$, we can multiply both sides of the equation by the denominator $\left(5 x^2+1\right)^2$ to eliminate the fractions. This gives us:

$$15 x^3+25 x^2+5 x+2 = A \left(5 x^2+1\right) + B$$

We can now expand the right-hand side of the equation and equate the coefficients of like terms.

Equating Coefficients

Equating the coefficients of like terms, we get:

$$15 x^3+25 x^2+5 x+2 = A \left(5 x^2+1\right) + B$$

$$15 x^3+25 x^2+5 x+2 = 5 A x^2 + A + B$$

Equating the coefficients of $x^3$, we get:

$$15 = 0$$

This implies that $A = 0$.

Equating the constant terms, we get:

$$2 = B$$

Therefore, we have found that $A = 0$ and $B = 2$.

Conclusion

In conclusion, the correct form of the partial fraction decomposition for the given expression $\frac{15 x^3+25 x^2+5 x+2}{\left(5 x^2+1\right)^2}$ is:

$$\frac{15 x^3+25 x^2+5 x+2}{\left(5 x^2+1\right)^2} = \frac{0}{5 x^2+1} + \frac{2}{\left(5 x^2+1\right)^2}$$

This is option B.

Final Answer

Introduction

Partial fraction decomposition is a powerful technique used in algebra to break down complex rational functions into simpler fractions. In our previous article, we explored the concept of partial fraction decomposition and applied it to the given expression $\frac{15 x^3+25 x^2+5 x+2}{\left(5 x^2+1\right)^2}$. In this article, we will answer some frequently asked questions about partial fraction decomposition.

Q: What is the purpose of partial fraction decomposition?

A: The purpose of partial fraction decomposition is to break down a complex rational function into simpler fractions, making it easier to analyze and solve equations.

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

A: You should use partial fraction decomposition when you have a rational function with a polynomial numerator and a polynomial denominator, and you want to express it as a sum of simpler fractions.

Q: What are the steps involved in partial fraction decomposition?

A: The steps involved in partial fraction decomposition are:

1. Factor the denominator: Factor the denominator of the rational function into its simplest form.
2. Write the partial fraction decomposition: Write the partial fraction decomposition as a sum of fractions, where each fraction has a polynomial numerator and a polynomial denominator.
3. Solve for the constants: Solve for the constants in the partial fraction decomposition by equating the coefficients of like terms.

Q: How do I solve for the constants in partial fraction decomposition?

A: To solve for the constants in partial fraction decomposition, you need to equate the coefficients of like terms. This involves multiplying both sides of the equation by the denominator and expanding the right-hand side.

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

A: Some common mistakes to avoid in partial fraction decomposition are:

• Not factoring the denominator: Make sure to factor the denominator into its simplest form.
• Not writing the partial fraction decomposition correctly: Make sure to write the partial fraction decomposition as a sum of fractions, where each fraction has a polynomial numerator and a polynomial denominator.
• Not solving for the constants correctly: Make sure to solve for the constants by equating the coefficients of like terms.

Q: Can I use partial fraction decomposition with rational functions that have repeated factors in the denominator?

A: Yes, you can use partial fraction decomposition with rational functions that have repeated factors in the denominator. However, you need to be careful when solving for the constants, as the repeated factors can lead to multiple solutions.

Q: How do I know if a rational function can be decomposed into partial fractions?

A: A rational function can be decomposed into partial fractions if and only if the denominator can be factored into linear factors. If the denominator cannot be factored into linear factors, then the rational function cannot be decomposed into partial fractions.

Conclusion

In conclusion, partial fraction decomposition is a powerful technique used in algebra to break down complex rational functions into simpler fractions. By following the steps involved in partial fraction decomposition and avoiding common mistakes, you can successfully decompose rational functions into partial fractions. We hope this Q&A guide has been helpful in answering your questions about partial fraction decomposition.

Final Tips

• Practice, practice, practice: The more you practice partial fraction decomposition, the more comfortable you will become with the technique.
• Use online resources: There are many online resources available that can help you with partial fraction decomposition, including video tutorials and practice problems.
• Seek help when needed: Don't be afraid to seek help if you are struggling with partial fraction decomposition. Your teacher or tutor can provide you with additional guidance and support.