Problem 41 - Precalculus: Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{x^{4}+x^{3}+x^{2}-x+1}{x\left(x^{2}+1\right)^{2}}$$

Step-by-Step Solution

Answer

Decomposition is achieved by solving the system for coefficients, resulting in distinct fractions for each term.

Step 1: Identify the Denominator

The denominator of the given rational function is \(x \left(x^2 + 1\right)^2\). It consists of a linear factor \(x\) and a repeated irreducible quadratic factor \(\left(x^2 + 1\right)^2\).

Step 2: Decompose into Partial Fractions

For the first linear factor \(x\), propose a term \(\frac{A}{x}\). For the repeated quadratic factor \(\left(x^2 + 1\right)^2\), include two terms: \(\frac{Bx + C}{x^2 + 1}\) and \(\frac{Dx + E}{(x^2 + 1)^2}\). Thus, the partial fraction decomposition is \[\frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{(x^2 + 1)^2}.\]

Step 3: Express in a Common Denominator

To express the decomposition in terms of a common denominator, set up the equation: \[ \frac{A(x^2 + 1)^2 + (Bx + C)x(x^2 + 1) + (Dx + E)x}{x(x^2 + 1)^2} = \frac{x^4 + x^3 + x^2 - x + 1}{x(x^2 + 1)^2}. \]

Step 4: Multiply and Simplify

Multiply out and simplify the numerator: \[ A(x^2 + 1)^2 + (Bx + C)x(x^2 + 1) + (Dx + E)x.\] Expand each term: \(A(x^4 + 2x^2 + 1)\), \((Bx + C)(x^3 + x)\), and \((Dx + E)x\). Combine the terms into a single polynomial in terms of the corresponding powers of \(x\).

Step 5: Solve for Coefficients using Equality of Polynomials

Since the polynomials in the numerators must be equal, equate corresponding coefficients of \(x^4\), \(x^3\), \(x^2\), \(x\), and the constant terms. Solve the resulting system of linear equations for the coefficients \(A\), \(B\), \(C\), \(D\), and \(E\).

Step 6: Write the Final Decomposition

Having solved the linear equations, substitute the values of \(A\), \(B\), \(C\), \(D\), and \(E\) into the partial fraction decomposition. Thus, the function decomposes as \(\frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{(x^2 + 1)^2}\) with specific calculated coefficients.

Key Concepts

Rational Functions

Rational functions are expressions created by dividing one polynomial by another. These functions are often seen in mathematics and engineering due to their manageable properties, even under complex conditions.
Understanding them can help solve numerous mathematical problems, especially those related to calculus and algebra.

In the given exercise, we start with the rational function \( \frac{x^{4}+x^{3}+x^{2}-x+1}{x(x^{2}+1)^{2}} \), where:

• The numerator is the polynomial \( x^{4}+x^{3}+x^{2}-x+1 \).
• The denominator is the polynomial \( x(x^{2}+1)^{2} \).

Rational functions embody the balance between two opposing forces: the numerator's power and the denominator's boundary. This interplay plays a crucial role in understanding limits, asymptotes, and behaviors of functions.

Linear Factors

Linear factors are the building blocks of polynomials, configured in a straightforward form \( ax + b \). They represent the simplest form of polynomial factors, making them crucial in simplifying complex expressions.
In the exercise, the factor \( x \) forms a linear component of the rational function's denominator. It's a fundamental ingredient in partial fraction decomposition. Linear factors can be easily managed by introducing simple fractions such as \( \frac{A}{x} \), where \( A \) is a constant determined through algebraic manipulation.

These factors are crucial when separating complex expressions into more approachable components, making calculations simpler and easing deeper analysis of polynomial behavior.

Irreducible Quadratic Factors

Irreducible quadratic factors are polynomial expressions of the form \( ax^2 + bx + c \) that cannot be simplified further over the real numbers. In partial fraction decomposition, identifying these factors is key to breaking down the rational function into more manageable parts.
In the exercise, the term \((x^2 + 1)\) is an irreducible quadratic factor repeated twice in the denominator. This requires considering it both as a simple and a squared factor, leading to terms \( \frac{Bx + C}{x^2 + 1} \) and \( \frac{Dx + E}{(x^2 + 1)^2} \), respectively.

This approach helps isolate different behaviours within rational functions, facilitating their analysis and resolution.

Common Denominator

The common denominator is an essential concept when combining fractions.For partial fractions' decomposition, aligning multiple fractions to a common denominator allows us to express the original rational function equivalently.Once fractions share a common base, equations can be equated by comparing numerators.

In our exercise, the goal is to express terms like \( \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{(x^2 + 1)^2} \) over the common denominator \( x(x^2 + 1)^2 \).Achieving this involves multiple steps of algebraic manipulation, requiring multiplication, simplification, and ultimately solving for unknown coefficients.
The practice of finding and equating via a common denominator not only assists in simplifying original expressions, but also enlightens the underlying relationships between polynomial components.