Problem 4 - 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. $$\frac{1}{x^{4}-x^{3}}$$

Step-by-Step Solution

Answer

\(\frac{1}{x^4 - x^3} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x - 1}\)

Step 1: Factoring the Denominator

The first step in partial fraction decomposition is to factor the denominator. The denominator here is \(x^4 - x^3\). Notice that we can factor out an \(x^3\) from the terms: \(x^4 - x^3 = x^3(x - 1)\). So the denominator as a product of factors is: \[x^3(x - 1)\].

Step 2: Identify the Types of Factors

Next, identify the types of factors. The term \(x^3\) is a repeated linear factor, while \(x - 1\) is a simple linear factor.

Step 3: Set Up the Partial Fractions

Write the form of the partial fraction decomposition. Since \(x^3\) is a repeated linear factor, it contributes three terms: \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3}\). The factor \(x - 1\) contributes one term, \(\frac{D}{x - 1}\). Thus, the partial fraction decomposition is: \[\frac{1}{x^4 - x^3} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x - 1}\].

Key Concepts

Factoring Polynomials

Factoring polynomials is a critical skill in algebra that involves breaking down a polynomial into simpler multiplicative terms, called factors. This simplification makes it easier to handle expressions and solve equations. In the given exercise, the polynomial was \( x^4 - x^3 \). To factor it, we look for common components in each term.

Here, both terms share \( x^3 \). By factoring out \( x^3 \), we simplify the polynomial as follows:

• Identify the common factor: \( x^3 \)
• Divide each term by \( x^3 \): \( x^4 - x^3 = x^3(x - 1) \)

This process of finding the greatest common factor and using it to simplify is fundamental in factoring polynomials. Once the polynomial is factored, it reveals the underlying structures that can later be decomposed into partial fractions.

Rational Expressions

A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Handling rational expressions requires understanding both the algebraic manipulation and simplification of these expressions. In the context of the exercise, our rational expression is \( \frac{1}{x^4 - x^3} \).

When working with rational expressions, consider the following steps:

• Identify and factor the denominator and numerator, if possible. This allows for simplification.
• Be cautious of restrictions. Since division by zero is undefined, identify values that make the denominator zero.
• Express the simplified form to aid in further analysis, such as partial fraction decomposition.

By approaching rational expressions methodically, you develop a deeper understanding and finesse in manipulating even the most complex algebraic expressions.

Linear Factors

In the realm of algebra, factors are categorized by their degree. Linear factors are first-degree polynomials that can take a simple form like \( x \) or \( x - 1 \). Recognizing linear factors is essential for procedures like partial fraction decomposition. From our example, the factors \( x^3 \) and \( x - 1 \) reveal the nature of such linear components.

Here’s how they play a role:

• \( x \) is a basic linear factor, repeated three times in \( x^3 \).
• \( x - 1 \) is another linear factor that appears only once.

Identifying these linear factors helps set up the partial fractions. Each unique linear factor contributes a term to the decomposition. When repeated, as in \( x^3 \), each power of the factor up to its total degree must be accounted for separately. Thus, we employ \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} \) for \( x^3 \), ensuring all possibilities are covered. Understanding linear factors means knowing their structure, influence, and how they relate to the broader equation.