Simplifying Fractions through Partial Fractions Decomposition: A Step-by-Step Guide

Partial fraction decomposition is a powerful technique used in algebra to simplify complex fractions into simpler, more manageable forms. In this article, we will walk through the process of simplifying the fraction (frac{2}{x^2 - 1}) using partial fractions. By understanding this method, you can tackle similar algebraic problems effectively.

Introduction to Partial Fractions

Partial fractions decomposition involves breaking down a single fraction with a complex denominator into simpler fractions. The denominator, in this case, is (x^2 - 1), which can be factored as ((x - 1)(x 1)). This step is crucial because it allows us to rewrite the fraction as the sum of simpler fractions whose denominators are the factors of the original denominator.

Step-by-Step Decomposition

Let's start by expressing (frac{2}{x^2 - 1}) in the form of partial fractions. We can write:

[frac{2}{x^2 - 1} frac{A}{x} frac{B}{x 1}]

To solve for (A) and (B), we will multiply both sides of the equation by the common denominator, which is (x^2 - 1):

[2 A(x 1) Bx]

We now have a linear system of equations to solve for (A) and (B). We can find these values by setting (x) to specific values that simplify the equation.

Setting (x 0)

When (x 0) in the equation:

[2 A(0 1) B(0)]

This simplifies to:

[2 A]

Therefore, (A 2).

Setting (x -1)

When (x -1) in the equation:

[2 A(-1 1) B(-1)]

This simplifies to:

[2 -B]

Therefore, (B -2).

Final Simplification

Substituting (A 2) and (B -2) back into the original partial fraction decomposition, we get:

[frac{2}{x^2 - 1} frac{2}{x} - frac{2}{x 1}]

Additional Method for Verification

Another method to verify this result is by rewriting the original fraction using algebraic manipulation. We can write:

[frac{2}{x^2 - 1} 2 left( frac{1}{x^2 - 1} right) 2 left( frac{1}{(x - 1)(x 1)} right)]

We can further decompose this as:

[2 left( frac{x 1 - (x - 1)}{(x - 1)(x 1)} right) 2 left( frac{x 1}{(x - 1)(x 1)} - frac{x - 1}{(x - 1)(x 1)} right) 2 left( frac{1}{x - 1} - frac{1}{x 1} right)]

Which simplifies to:

[frac{2}{x^2 - 1} frac{2}{x} - frac{2}{x 1}]

Conclusion

Partial fraction decomposition is a powerful tool in algebra, particularly when dealing with complex fractions. The process we have demonstrated here helps to break down a complex fraction into simpler, more manageable parts. This technique is widely used in various mathematical fields, including calculus, physics, and engineering.

By following the steps outlined in this guide, you can simplify similar fractions with confidence. Understanding and mastering partial fraction decomposition can greatly enhance your problem-solving skills in mathematics and related disciplines.