Problem 6 - Algebra and Trigonometry

Question

\(3-12\) . 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

The partial fraction decomposition is \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x-1}\).

Step 1: Factor the Denominator

The first step is to factor the denominator of the given function. The denominator is \(x^4 - x^3\). Factor out the greatest common factor, which is \(x^3\), resulting in: \[ x^4 - x^3 = x^3(x-1) \]

Step 2: Set Up Partial Fraction Decomposition

Given the factored form \(x^3(x-1)\), write the expression for partial fractions. Since \(x^3\) is a repeated factor, each form (\(x\), \(x^2\), \(x^3\)) needs a separate term. The factor \(x-1\) is a simple linear factor. Therefore, the partial fraction decomposition is:\[ \frac{1}{x^3(x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x-1} \]

Key Concepts

Rational Functions

Rational functions are a type of function that represent the ratio of two polynomials. These functions can be expressed in the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)eq0\). This formula defines a relationship between two polynomial expressions, indicating how one changes in response to changes in the other.
It's important to understand that the behavior of rational functions is dictated by their numerators and denominators.

• The numerator \(P(x)\) controls the zeros of the function, or where the function equals zero.
• The denominator \(Q(x)\) determines the function's asymptotes, or where the function is undefined (these are the *singularities*).

When dealing with rational functions, a common task is to simplify or rewrite them using techniques such as partial fraction decomposition. This is particularly useful in calculus when integrating rational functions.

Factoring Polynomials

Factoring polynomials is a crucial skill for determining a rational function's behavior. It involves breaking down a polynomial into simpler polynomials that, when multiplied together, produce the original polynomial.
For example, in the provided problem, the denominator \(x^4 - x^3\) is factored by pulling out the greatest common factor, which is \(x^3\). This simplifies the problem significantly, transforming it into \(x^3(x-1)\).
Understanding how to factor polynomials effectively helps in:

• Finding the zeros of the polynomial, which correspond to the function's x-intercepts or roots.
• Simplifying the rational function for easier integration or differentiation.
• Preparing the function for partial fraction decomposition by identifying repeated factors and handling each separately.

Once factored, polynomials offer clearer insight into their structure, which aids in the further analysis of rational functions and makes them easier to work with.

Repeated Factors

Repeated factors in polynomials are factors that appear more than once. In the context of partial fraction decomposition, it's important to address each of these factors separately.
In our exercise, \(x^3\) is a repeated factor, which means we need to account for each power of \(x\) up to the highest occurrence, in this case, \(x^3\). This results in individual terms for each occurrence:

• \(\frac{A}{x}\)
• \(\frac{B}{x^2}\)
• \(\frac{C}{x^3}\)

Each term represents a different part of the behavior of the original rational function. By separately identifying these terms, we can more accurately perform operations such as integrations.
Repeated factors can increase the algebraic complexity, but by breaking them down, we transform the rational function into more manageable parts, thus simplifying further mathematical operations.