Expressing the Rational Function in Partial Fractions

Oct 13, 2025 by magicalceylontours.com 39 views

Introduction

Partial fraction decomposition is a powerful technique used in algebra to simplify complex rational functions into more manageable components. In this article, we will explore the partial fraction decomposition of the rational function $$\frac{52}{x^2-4}$$ and express it in the form of $$\frac{f(x)}{x-2}+\frac{g(x)}{x+2}$$, where $f(x)$ and $g(x)$ are polynomials to be determined.

What is Partial Fraction Decomposition?

Partial fraction decomposition is a method of expressing a rational function as a sum of simpler fractions. It is a powerful tool used in algebra to simplify complex rational functions and make them easier to work with. The process of partial fraction decomposition involves expressing a rational function as a sum of simpler fractions, each with a polynomial numerator and a linear or quadratic denominator.

The Rational Function

The rational function we will be working with is $$\frac{52}{x^2-4}$$. This function can be factored as $$\frac{52}{(x-2)(x+2)}$$, where $x^2-4$ is the denominator and $(x-2)(x+2)$ is the factorization of the denominator.

Expressing the Rational Function in Partial Fractions

To express the rational function in partial fractions, we need to find two polynomials $f(x)$ and $g(x)$ such that $$\frac{52}{x^2-4} = \frac{f(x)}{x-2} + \frac{g(x)}{x+2}$$. To do this, we can use the method of equating coefficients.

Equating Coefficients

We can start by multiplying both sides of the equation by the common denominator $(x-2)(x+2)$ to get rid of the fractions. This gives us $$52 = f(x)(x+2) + g(x)(x-2)$$.

Finding the Polynomials

To find the polynomials $f(x)$ and $g(x)$ , we can use the method of equating coefficients. We can expand the right-hand side of the equation and equate the coefficients of the corresponding terms on both sides.

Expanding the Right-Hand Side

Expanding the right-hand side of the equation gives us $$52 = f(x)(x+2) + g(x)(x-2) = (f(x)x + 2f(x)) + (g(x)x - 2g(x))$$.

Equating Coefficients

Equating the coefficients of the corresponding terms on both sides gives us the following system of equations:

$f(x)x + 2f(x) = 52$
$g(x)x - 2g(x) = 0$

Solving the System of Equations

To solve the system of equations, we can start by solving the second equation for $g(x)$ . This gives us $$g(x) = \frac{2}{x-2}$$.

Substituting into the First Equation

Substituting $g(x) = \frac{2}{x-2}$ into the first equation gives us $$f(x)x + 2f(x) = 52$$.

Solving for $f(x)$

Solving for $f(x)$ gives us $$f(x) = \frac{26}{x+2}$$.

Conclusion

In this article, we have used the method of partial fraction decomposition to express the rational function $$\frac{52}{x^2-4}$$ in the form of $$\frac{f(x)}{x-2}+\frac{g(x)}{x+2}$$. We have found that $f(x) = \frac{26}{x+2}$ and $g(x) = \frac{2}{x-2}$ .

Applications of Partial Fraction Decomposition

Partial fraction decomposition has many applications in algebra and calculus. It is used to simplify complex rational functions, make them easier to work with, and solve equations. It is also used to find the inverse of a matrix and to solve systems of linear equations.

Real-World Applications

Partial fraction decomposition has many real-world applications. It is used in engineering to design and analyze electrical circuits, in physics to solve problems involving motion and forces, and in economics to model and analyze economic systems.

Conclusion

Q: What is partial fraction decomposition?

A: Partial fraction decomposition is a method of expressing a rational function as a sum of simpler fractions. It is a powerful tool used in algebra to simplify complex rational functions and make them easier to work with.

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 or quadratic factors.

Q: What are the steps to decompose a rational function into partial fractions?

A: The steps to decompose a rational function into partial fractions are:

Factor the denominator into linear or quadratic factors.
Write the rational function as a sum of simpler fractions, each with a polynomial numerator and a linear or quadratic denominator.
Equate the coefficients of the corresponding terms on both sides.
Solve the resulting system of equations to find the polynomials.

Q: How do I find the polynomials in a partial fraction decomposition?

A: To find the polynomials in a partial fraction decomposition, you can use the method of equating coefficients. You can expand the right-hand side of the equation and equate the coefficients of the corresponding terms on both sides.

Q: What are some common mistakes to avoid when decomposing a rational function into partial fractions?

A: Some common mistakes to avoid when decomposing a rational function into partial fractions include:

Not factoring the denominator correctly.
Not writing the rational function as a sum of simpler fractions.
Not equating the coefficients correctly.
Not solving the resulting system of equations correctly.

Q: How do I check my work when decomposing a rational function into partial fractions?

A: To check your work when decomposing a rational function into partial fractions, you can:

Multiply both sides of the equation by the common denominator.
Expand the right-hand side of the equation.
Equate the coefficients of the corresponding terms on both sides.
Solve the resulting system of equations.

Q: What are some real-world applications of partial fraction decomposition?

A: Some real-world applications of partial fraction decomposition include:

Designing and analyzing electrical circuits.
Solving problems involving motion and forces.
Modeling and analyzing economic systems.
Finding the inverse of a matrix and solving systems of linear equations.

Q: Can partial fraction decomposition be used to solve differential equations?

A: Yes, partial fraction decomposition can be used to solve differential equations. It can be used to simplify complex rational functions and make them easier to work with.

Q: Are there any limitations to partial fraction decomposition?

A: Yes, there are some limitations to partial fraction decomposition. It can only be used to decompose rational functions with linear or quadratic denominators. It cannot be used to decompose rational functions with polynomial denominators of degree greater than 2.

Q: Can partial fraction decomposition be used to find the inverse of a matrix?

A: Yes, partial fraction decomposition can be used to find the inverse of a matrix. It can be used to simplify complex rational functions and make them easier to work with.

Conclusion

In conclusion, partial fraction decomposition is a powerful technique used in algebra to simplify complex rational functions. It is used to express a rational function as a sum of simpler fractions, each with a polynomial numerator and a linear or quadratic denominator. It has many applications in algebra and calculus, and is used to solve equations, find the inverse of a matrix, and solve systems of linear equations.