Problem 4 - Essential Calculus

Question

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-1}{x\left(x^{2}+1\right)^{2}} $$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational expression, without solving for the coefficients, can be written as \(\frac{a}{x} + \frac{bx + c}{x^2+1} + \frac{dx + e}{(x^2+1)^2}\).

Step 1: Identifying the General Form

For the given rational expression, \(\frac{2x - 1}{x(x^2+1)^2}\), we first need to identify the general form of the partial fractions decomposition. The distinct factors are \(x\) and \((x^2 + 1)\). Since \((x^2 + 1)\) appears twice in the denominator, we account for it twice in the decomposition. Therefore, our general form will look like: A/x + B/(x^2+1) + C/(x^2+1)^2

Step 2: Expressing the Coefficients as Variables

Next, to accommodate the degree of \(x\) in the numerator, we generalize the numerators of the fractions with polynomial variables. We shift from constants (\(A, B, C\)) to variables (\(a, bx+c, dx+e\)). Now the desired form of partial fraction decomposition for the given equation will be:\n\[\frac{a}{x} + \frac{bx + c}{x^2+1} + \frac{dx + e}{(x^2+1)^2}\]

Key Concepts

Understanding Rational Expressions

A rational expression is simply a fraction where both the numerator and the denominator are polynomials. Just as with ordinary fractions consisting of integers, rational expressions express a ratio between two mathematical expressions. They can often be simplified or decomposed into simpler parts.

Let's break it down further:

• Numerator: The top part of the fraction, which is a polynomial. In our exercise, it is \(2x - 1\).
• Denominator: The bottom part of the fraction. For our task, it is made up of the polynomial factors \(x(x^2 + 1)^2\).

These expressions can be manipulated and rewritten to make them easier to understand or to operate with, such as in integration or solving equations. Writing a rational expression as a sum of simpler fractions is known as partial fraction decomposition.

The Role of Polynomials

Polynomials are fundamental building blocks in mathematics and are crucial when dealing with rational expressions. A polynomial is an expression formed by summing constants and variables raised to natural number powers, ordered by degree.

In the expression \(2x - 1\), \(x\) is the variable, and the highest exponent (or power) tells us about the degree of the polynomial. Here it's a linear polynomial because the highest power of \(x\) is one.

In the decomposition process:

• We see simple polynomials like \(x\).
• We encounter more complex quadratic polynomials, such as \(x^2 + 1\), appearing twice in the denominator.

Different powers of the polynomial in the denominator impact how we express the individual terms in the decomposition, leading us into the realm of denominator factors.

Exploring Denominator Factors

In rational expressions, the denominator's factors heavily influence the form of partial fraction decompositions. A factor is a simpler polynomial multiplied to form part of the bigger polynomial (our entire denominator). The expression \(x(x^2 + 1)^2\) has foundational factors that guide our decomposition.

For partial fraction decomposition:

• For the factor \(x\), we use a single fraction \(\frac{a}{x}\).
• For \(x^2 + 1\), which appears twice, we add the fractions \(\frac{bx + c}{x^2+1}\) and \(\frac{dx + e}{(x^2+1)^2}\).

This treatment allows us to match the identity of the original rational expression. Each factor contributes to the system of equations we would eventually solve for specific coefficients. This methodical breakdown highlights the power of recognizing and utilizing denominator factors in decomposing rational expressions.