Problem 12 - Precalculus Mathematics 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. $$\frac{1}{\left(x^{3}-1\right)\left(x^{2}-1\right)}$$

Step-by-Step Solution

Answer

\[ \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{Cx+F}{x^2+x+1} + \frac{D}{x+1} \]

Step 1: Factor the Denominator

First, we need to factor the denominator \( (x^3-1)(x^2-1) \). The expression \( x^3-1 \) can be factored as \( (x-1)(x^2+x+1) \), and \( x^2-1 \) can be factored as \( (x-1)(x+1) \). Therefore, the overall factorization is \( (x-1)^2(x^2+x+1)(x+1) \).

Step 2: Set Up the Partial Fraction Form

With the factorization from Step 1, set up the partial fraction decomposition. Each linear factor \( (x-1) \) and \( (x+1) \) will have a constant \( A \) and \( D \) as a coefficient: \( \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{D}{x+1} \). The irreducible quadratic factor, \( x^2+x+1 \), requires a linear numerator: \( \frac{Cx+F}{x^2+x+1} \). Consequently, the decomposition form is:\[ \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{Cx+F}{x^2+x+1} + \frac{D}{x+1} \].

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. Factors are simpler polynomials that, when multiplied together, give back the original polynomial. This technique is essential in simplifying expressions and solving polynomial equations. By understanding polynomial factorization, you're essentially breaking down complex problems into manageable pieces.

For example, in the problem where we consider the denominator \((x^3-1)(x^2-1)\), each polynomial can be decomposed into smaller factors:

• \(x^3-1\) factors to \((x-1)(x^2+x+1)\)
• \(x^2-1\) factors to \((x-1)(x+1)\)

In our example, this results in \((x-1)^2(x^2+x+1)(x+1)\), a product of two linear factors and one irreducible quadratic factor. Understanding how to factor polynomials is crucial for tackling more advanced problems in calculus and algebra. This will help you in setting up partial fraction decompositions, solving equations, and within various applications in mathematics.

Rational Functions

Rational functions are functions of the form \( \frac{p(x)}{q(x)} \), where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) eq 0\). They are an essential part of algebra and calculus as they help describe a wide variety of phenomena, from physics to economics.

In our partial fraction decomposition example, the rational function is \( \frac{1}{(x^3-1)(x^2-1)} \). Here, the numerator is just 1, and the denominator is the product of two polynomials.

Working with rational functions often involves simplifying, factoring, and using partial fraction decomposition to make calculus operations like integration more manageable. These skills are especially useful in finding antiderivatives of complex functions, analyzing behavior near asymptotes, and understanding the overall shape and behavior of graphs of these functions.

Algebraic Fractions

Algebraic fractions are similar to regular fractions but involve polynomials. Just like with numerical fractions, simplifying or decomposing algebraic fractions can make them easier to work with. This often involves factorization of the numerator and denominator, cancellation of common factors, and setting up partial fraction forms.

When dealing with complex algebraic fractions, as seen in the example \( \frac{1}{(x^3-1)(x^2-1)} \), using polynomial factorization allows us to break the fraction into more manageable parts by applying partial fraction decomposition. The aim is to express the function as a sum of simpler fractions:

• Each simpler fraction aligns with a component from the factored form of the denominator.
• The process involves assigning constants or linear terms (e.g., \( A, B, Cx+F, D \)) appropriate to these components.

This transformation facilitates solving equations, integrating functions, and comprehending the proportions and relationships within algebraic systems. Grasping algebraic fractions is a cornerstone skill in higher-level mathematics that can optimize calculations and boost conceptual understanding across diverse mathematical contexts.