Problem 30 - Algebra and Trigonometry

Question

Find the partial fraction decomposition of the rational function. \(\frac{x^{3}-2 x^{2}-4 x+3}{x^{4}}\)

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{1}{x} - \frac{2}{x^2} - \frac{4}{x^3} + \frac{3}{x^4}\).

Step 1: Identify the type of fraction

The given fraction is \(\frac{x^3 - 2x^2 - 4x + 3}{x^4}\), where the denominator is a single polynomial term \(x^4\). This indicates that we are working with a proper fraction.

Step 2: Express the rational function as a sum of partial fractions

Since the denominator is \(x^4\), we express the function as \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4}\). Here, \(A\), \(B\), \(C\), and \(D\) are constants that we need to determine.

Step 3: Rewrite as a single fraction

Combine the right-hand side to rewrite it as a single fraction: \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} = \frac{Ax^3 + Bx^2 + Cx + D}{x^4}\). Set this equal to the original function \(\frac{x^3 - 2x^2 - 4x + 3}{x^4}\).

Step 4: Compare coefficients

Since the denominators are the same, compare the numerators: \(Ax^3 + Bx^2 + Cx + D = x^3 - 2x^2 - 4x + 3\). This gives us a system of equations: \(A = 1\), \(B = -2\), \(C = -4\), \(D = 3\).

Step 5: Solve for constants

From the above system of equations, we directly have the solutions: \(A = 1\), \(B = -2\), \(C = -4\), and \(D = 3\).

Step 6: Write the partial fraction decomposition

Substitute the constants back into the partial fractions to get the decomposition: \(\frac{1}{x} - \frac{2}{x^2} - \frac{4}{x^3} + \frac{3}{x^4}\).

Key Concepts

Rational Functions

Rational functions are an essential concept in algebra. They are formed by dividing two polynomials. Specifically, they take the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). This concept is pivotal for understanding more complex mathematical operations, such as partial fraction decomposition.

A key aspect of rational functions is their behavior at various points, especially at the roots of the denominator. These are points of discontinuity, commonly known as vertical asymptotes. Rational functions are useful for modeling situations where two quantities depend on each other and can be represented by polynomial expressions.

Examples of rational functions include fractions like \( \frac{2x+1}{x-3} \) or more complex expressions like \( \frac{x^2 - 4x + 4}{x^2 - 1} \). Understanding the nature of both the numerator and the denominator can give you insights into the graph and behavior of the function.

Proper Fraction

In the context of rational functions, a proper fraction has a numerator of lesser degree than the denominator. This is crucial when performing partial fraction decomposition, a technique used to break complex fractions into simpler parts.

For instance, in the given exercise, the rational function \( \frac{x^3 - 2x^2 - 4x + 3}{x^4} \) is considered a proper fraction because the degree of the numerator (3) is less than the degree of the denominator (4). This relationship dictates the decomposition process.

A proper fraction ensures that the terms can be broken down into simpler components without increasing the complexity of the function. In improper fractions, additional steps, such as polynomial division, might be necessary to convert them into the form suitable for decomposition.

Coefficients

Coefficients are constants that define the leading terms in a polynomial expression. When breaking down a rational function through partial fraction decomposition, determining the coefficients of each term in the fraction is a crucial step.

A typical approach is to express the rational function as a sum of simpler fractions, as shown in the step-by-step solution. For example, \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} \) requires finding the coefficients \( A, B, C, \) and \( D \).

To find these coefficients, set up a system of equations by equating the original function's numerator with the expanded evaluator's numerator. Here, the exercise involves solving \( A = 1 \), \( B = -2 \), \( C = -4 \), and \( D = 3 \). These coefficients serve as constants in the resulting decomposed fractions, illustrating the essential role they play in restructuring and simplifying complex expressions.