Problem 8 - Algebra and Trigonometry

Question

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. \(\frac{x^{4}+x^{2}+1}{x^{2}\left(x^{2}+4\right)^{2}}\)

Step-by-Step Solution

Answer

\( \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 4} + \frac{Ex + F}{(x^2 + 4)^2} \)

Step 1: Identify the Denominator

The first step in solving a partial fraction decomposition is to analyze the denominator of the rational function. We have: \( x^2 (x^2 + 4)^2 \). Notice that it consists of a repeated irreducible quadratic factor \((x^2 + 4)^2\) and a simple linear factor \(x^2\) designating multiplicity.

Step 2: Set Up Partial Fractions Structure

Based on the denominator, we set up the structure of the partial fraction decomposition. Each linear factor \(x^2\) of multiplicity 2 will correspond to the terms \(\frac{A}{x} + \frac{B}{x^2}\). The irreducible quadratic \((x^2 + 4)^2\) will correspond to the terms \(\frac{Cx + D}{x^2 + 4} + \frac{Ex + F}{(x^2 + 4)^2}\).

Step 3: Write the Partial Fraction Decomposition

Combining these terms, the partial fraction decomposition of the given function will be:\[\frac{x^4 + x^2 + 1}{x^2(x^2 + 4)^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 4} + \frac{Ex + F}{(x^2 + 4)^2}\]

Key Concepts

Rational Functions

Rational functions are a fundamental concept in algebra, critical for understanding partial fraction decomposition. These functions are expressed as the quotient of two polynomials, where the numerator and the denominator are polynomials. In the fractional form, \[ \frac{P(x)}{Q(x)} \]- \(P(x)\) stands for the numerator polynomial, and - \(Q(x)\) stands for the denominator polynomial.
The rational function is essentially about division and creating quotients. Partial fraction decomposition, the focus here, manipulates these functions to express them as sums of simpler fractions. This is particularly useful in calculus, especially in solving integrals involving rational functions. Understanding the structure of these functions is key to breaking them down into partial fractions.

Irreducible Quadratic Factor

An essential aspect of partial fraction decomposition is recognizing irreducible quadratic factors within a rational function's denominator. An irreducible quadratic factor is a quadratic expression that cannot be factored further over the real numbers. For example, \(x^2 + 4\) is irreducible because it doesn't have real roots. In other words, there is no solution to the equation \(x^2 + 4 = 0\) in the set of real numbers.
When encountering such factors, they typically contribute to a more complex partial fraction setup, involving terms of the form \(\frac{Cx + D}{x^2 + 4}\).
A good understanding of these factors ensures the correct structure when decomposing rational functions. It's crucial when dealing with repeated or multiple occurrences of these terms, which require additional fractions.

Multiplicity

Multiplicity refers to how many times a specific factor appears in the factorization of a polynomial within the denominator of a rational function. If a factor occurs more than once, it is considered to have a multiplicity greater than one.
In the problem, the factor \((x^2 + 4)^2\) has a multiplicity of 2 because it appears squared. This indicates that you need to create separate terms in the partial fraction decomposition for each repetition, considering all levels of multiplicity.
For a factor like \(x^2\), the terms would be \(\frac{A}{x} + \frac{B}{x^2}\), properly handling its multiplicity of 2.- Understanding multiplicity is pivotal to creating accurate and complete partial fraction decompositions.

Numerical Coefficients

Numerical coefficients are the constants that accompany variables in mathematical expressions. In partial fraction decomposition, these coefficients are represented by letters like \(A, B, C, D\), etc.
These letters indicate the unknowns or constants that need to be determined to fully complete the decomposition. Although in our problem we are not tasked with finding these coefficients, understanding their role is important.
The use of numerical coefficients helps in expressing each simple fraction in the decomposition accurately. These coefficients are found by equating and solving equations derived from setting the rational function equal to its decomposed parts.- Coefficients help customize and tailor the decomposition to fit the original rational function perfectly.