The Correct Form of the Partial Fraction Decomposition

Oct 13, 2025 by talynt.org 98 views

What is the Correct Form of the Partial Fraction Decomposition for $\frac{6x+7}{(x+2)^2}$ ?

Partial fraction decomposition is a mathematical technique used to break down a rational function into simpler fractions. It is a powerful tool for solving algebraic equations and simplifying complex expressions. In this article, we will explore the correct form of the partial fraction decomposition for the rational function $\frac{6x+7}{(x+2)^2}$ .

Partial fraction decomposition involves expressing a rational function as a sum of simpler fractions. The general form of a partial fraction decomposition is:

\frac{P(x)}{Q(x)} = \frac{A_1}{(x-a_1)} + \frac{A_2}{(x-a_2)} + \cdots + \frac{A_n}{(x-a_n)}

where $P(x)$ and $Q(x)$ are polynomials, and $A_1, A_2, \ldots, A_n$ are constants.

The Rational Function $\frac{6x+7}{(x+2)^2}$

The rational function $\frac{6x+7}{(x+2)^2}$ has a numerator of degree 1 and a denominator of degree 2. To decompose this function into partial fractions, we need to find the constants $A$ and $B$ such that:

\frac{6x+7}{(x+2)^2} = \frac{A}{x+2} + \frac{B}{(x+2)^2}

Finding the Constants $A$ and $B$

To find the constants $A$ and $B$ , we can use the method of equating coefficients. We multiply both sides of the equation by $(x+2)^2$ to get:

6x+7 = A(x+2) + B

We can expand the right-hand side of the equation to get:

6x+7 = Ax + 2A + B

We can equate the coefficients of the $x$ terms on both sides of the equation to get:

6 = A

We can equate the constant terms on both sides of the equation to get:

7 = 2A + B

Substituting $A=6$ into the second equation, we get:

7 = 12 + B

Solving for $B$ , we get:

B = -5

The Correct Form of the Partial Fraction Decomposition

Therefore, the correct form of the partial fraction decomposition for the rational function $\frac{6x+7}{(x+2)^2}$ is:

\frac{6x+7}{(x+2)^2} = \frac{6}{x+2} - \frac{5}{(x+2)^2}

In this article, we have explored the correct form of the partial fraction decomposition for the rational function $\frac{6x+7}{(x+2)^2}$ . We have used the method of equating coefficients to find the constants $A$ and $B$ and have shown that the correct form of the partial fraction decomposition is $\frac{6}{x+2} - \frac{5}{(x+2)^2}$ .

The correct answer is A. $\frac{A}{x+2} + \frac{B}{(x+2)^2}$ , where $A=6$ and $B=-5$ .
What is the Correct Form of the Partial Fraction Decomposition for $\frac{6x+7}{(x+2)^2}$ ? A Q&A Article

In our previous article, we explored the correct form of the partial fraction decomposition for the rational function $\frac{6x+7}{(x+2)^2}$ . In this article, we will answer some frequently asked questions about partial fraction decomposition and provide additional examples to help you understand the concept better.

Q: What is partial fraction decomposition?

A: Partial fraction decomposition is a mathematical technique used to break down a rational function into simpler fractions. It is a powerful tool for solving algebraic equations and simplifying complex expressions.

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 degree of the numerator is less than the degree of the denominator.

Q: What is the general form of a partial fraction decomposition?

A: The general form of a partial fraction decomposition is:

\frac{P(x)}{Q(x)} = \frac{A_1}{(x-a_1)} + \frac{A_2}{(x-a_2)} + \cdots + \frac{A_n}{(x-a_n)}

where $P(x)$ and $Q(x)$ are polynomials, and $A_1, A_2, \ldots, A_n$ are constants.

Q: How do I find the constants $A$ and $B$ in a partial fraction decomposition?

A: To find the constants $A$ and $B$ , you can use the method of equating coefficients. You multiply both sides of the equation by the denominator to get:

P(x) = A(x-a_1) + B(x-a_2) + \cdots + C(x-a_n)

You can then equate the coefficients of the $x$ terms on both sides of the equation to get:

A = \frac{P(x)}{(x-a_1)}

You can also equate the constant terms on both sides of the equation to get:

B = \frac{P(x)}{(x-a_2)}

Q: What is the correct form of the partial fraction decomposition for $\frac{6x+7}{(x+2)^2}$ ?

A: The correct form of the partial fraction decomposition for $\frac{6x+7}{(x+2)^2}$ is:

\frac{6x+7}{(x+2)^2} = \frac{6}{x+2} - \frac{5}{(x+2)^2}

Q: Can you provide more examples of partial fraction decomposition?

A: Yes, here are a few more examples:

$\frac{x+1}{x^2-4} = \frac{1}{x-2} + \frac{1}{x+2}$
$\frac{x^2+3x+2}{x^2-4} = \frac{x+1}{x-2} + \frac{2}{x+2}$
$\frac{x^2+2x-3}{x^2-4} = \frac{x-1}{x-2} + \frac{2}{x+2}$

In this article, we have answered some frequently asked questions about partial fraction decomposition and provided additional examples to help you understand the concept better. We hope that this article has been helpful in clarifying any doubts you may have had about partial fraction decomposition.

If you are interested in learning more about partial fraction decomposition, we recommend checking out the following resources:

Khan Academy: Partial Fractions
MIT OpenCourseWare: Partial Fractions
Wolfram MathWorld: Partial Fractions

We hope that this article has been helpful in your understanding of partial fraction decomposition. If you have any further questions or need additional help, please don't hesitate to ask.