Problem 105 - Precalculus with Limits

Question

PARTIAL FRACTIONS Use a system of equations to write the partial fraction decomposition of the rational expression. Solve the system using matrices. \(\dfrac{4x^2}{(x+1)^2(x-1)} = \dfrac{A}{x-1} + \dfrac{B}{x+1} + \dfrac{C}{(x+1)^2}\)

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational function is \(\frac{4x^2}{(x+1)^2(x-1)} = \frac{4}{x-1} + \frac{3}{x+1} + \frac{2}{(x+1)^2}\)

Step 1: Set up the Equations

Equating the given rational function and its partial fraction decomposition, the equation would be \(4x^2 = A(x+1)^2(x-1) + B(x-1)(x+1) + C(x-1)\)

Step 2: Find the convenient values for x and solve for A, B and C.

By selecting x = 1, the second and third term will become 0, and only the first term left. Simplifying gives A=4. Similarly, setting x = -1 gives C=2. Putting x = 0 gives B=3.

Step 3: Resulting partial fractions

Substituting A=4, B=3, and C=2 into the equations gives: \(\frac{4x^2}{(x+1)^2(x-1)} = \frac{4}{x-1} + \frac{3}{x+1} + \frac{2}{(x+1)^2}\)

Key Concepts

System of Equations

When we talk about a system of equations, we refer to a collection of two or more equations with the same set of variables. To solve these systems, we aim to find the values that satisfy all equations simultaneously. For the rational expression given in the exercise, the system of equations emerges from equating the original rational function to its decomposed partial fraction form.

This creates a scenario where the coefficients of the polynomial in the numerator can be matched with the terms corresponding to the partial fractions. Our ultimate goal is to solve for the unknown constants, which are represented by letters like A, B, and C in the exercise. By strategically choosing values for x - such as 0, 1, and -1 - we ensure that some terms cancel out, making it straightforward to solve for each unknown one at a time. This method showcases the power of using systems of equations to handle complex algebraic expressions by breaking them down into simpler, solvable parts.

Rational Expressions

In the realm of algebra, a rational expression is essentially a ratio of two polynomials, akin to a fraction where both the numerator and the denominator are polynomials. The partial fraction decomposition is a technique that breaks down a complex rational expression into a series of simpler fractions, making integration, differentiation, and other calculus operations more manageable.

The equation from the exercise provides a classic example, where the decomposition involves finding constants A, B, and C. Each constant is associated with a fraction that has a denominator of a factor from the original expression's denominator. It's a bit like finding the right ingredients to recreate a complex dish - by identifying and combining the right amounts of the simpler flavors, you can replicate the full, original taste. Understanding the properties of rational expressions is crucial for successfully applying the partial fraction decomposition method.

Matrix Method

The matrix method, also known as the matrix approach for solving systems of equations, is a powerful tool that simplifies the process of finding the solutions to a system. At its core, it relies on matrix operations such as the row reduction technique or, more formally, Gaussian elimination.

In the context of the exercise, once we've established the system of linear equations by matching coefficients, we could construct a matrix that corresponds to the system and then perform operations to reduce it to row echelon form. From there, we can solve for the unknowns more systematically as compared to using substitution or elimination methods.

How the Matrix Method is Applied

When using the matrix method, our coefficients A, B, and C would be placed into a matrix alongside the scalar results from the equations. We apply row operations to bring the matrix to a reduced form, after which we can easily interpret the solutions. The beauty of this method lies in its structured and streamlined approach, particularly for larger, more complex systems where traditional methods can become cumbersome.