Problem 7 - Precalculus: Mathematical for Calculus

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. $$\frac{x^{3}-4 x^{2}+2}{\left(x^{2}+1\right)\left(x^{2}+2\right)}$$

Step-by-Step Solution

Answer

Express as: \(\frac{Ax+B}{x^2+1} + \frac{Cx+D}{x^2+2}\).

Step 1: Understanding Partial Fraction Decomposition

Partial fraction decomposition aims to express a rational function as a sum of simpler fractions. For this, ensure the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division before proceeding.

Step 2: Express the Denominator Components

Identify the factors in the denominator. Here, the denominator is \((x^2+1)(x^2+2)\), with distinct irreducible quadratic factors \(x^2+1\) and \(x^2+2\).

Step 3: Formulate the Partial Fraction Decomposition

Each quadratic factor in the denominator will have a corresponding fraction with a linear numerator. Thus, \(\frac{x^3-4x^2+2}{(x^2+1)(x^2+2)}\) can be expressed as \(\frac{Ax+B}{x^2+1} + \frac{Cx+D}{x^2+2}\), where \(A, B, C,\) and \(D\) are constants to be determined.

Key Concepts

Polynomial Long Division

Polynomial long division is a technique used to divide one polynomial by another when the degree of the numerator is greater than or equal to the degree of the denominator. It works similarly to arithmetic long division. This method can transform a rational function into a simpler form necessary for further analysis, such as partial fraction decomposition.

In some cases, a rational function might need a polynomial long division if the numerator's degree is not less than the denominator's degree. By performing long division, you can bring the rational function into a suitable form, where the degree of the numerator is less than the degree of the denominator.

For example, given a rational function like \(\frac{x^3 - 4x^2 + 2}{(x^2+1)(x^2+2)}\), if polynomial long division is required, it would allow us to separate the polynomial expression into simpler parts. These parts might be a polynomial and a rational function with a lower degree numerator.

Irreducible Quadratic Factors

Irreducible quadratic factors are quadratic expressions that cannot be factored into real linear factors. In the context of partial fraction decomposition, these factors are crucial because each irreducible factor in the denominator will correspond to a specific format in the decomposition.

In the given exercise, the denominator \( (x^2+1)(x^2+2) \) consists of two irreducible quadratic factors: \(x^2+1\) and \(x^2+2\). These factors cannot be simplified further using real numbers, thus the importance of considering them separately in partial fraction decomposition.

When constructing the partial fraction decomposition for each irreducible quadratic factor, you assign a linear expression in the numerator. For instance, \(\frac{Ax+B}{x^2+1}\) represents the decomposition over the factor \(x^2+1\), where \(A\) and \(B\) are constants.

Understanding how to handle these factors helps in rewriting complex rational functions into simpler forms, facilitating integration or interpretation.

Rational Functions

A rational function is a ratio of two polynomials. In the format \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not zero. These functions often appear in mathematical analysis and can vary from simple fractions to complex expressions involving polynomial equations of a higher degree.

The rational function in this exercise is \(\frac{x^3-4x^2+2}{(x^2+1)(x^2+2)}\). The main goal is often to simplify these functions and understand their properties by breaking them into simpler fractions, known as partial fractions.

Simplifying complex rational functions involves identifying and working with the factors of the denominator, such as irreducible quadratic factors. This allows solving, integrating, or applying other operations more easily, turning complex algebraic problems into manageable tasks. Mastering the concept of rational functions and their decomposition plays a significant role in calculus and algebra.