Problem 30 - Precalculus: Mathematical for Calculus

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

Step 1: Identify the Form of the Partial Fraction

The given rational function is \(\frac{x^3 - 2x^2 - 4x + 3}{x^4}\). Since the denominator is \(x^4\), it implies that there are repeated linear factors of \(x\) with multiplicities. Therefore, the partial fraction decomposition will have the form:\[\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4}\]where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Step 2: Write the Decomposition Equation

Set up the equation for the partial fraction decomposition:\[\frac{x^3 - 2x^2 - 4x + 3}{x^4} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4}\]Multiply both sides by \(x^4\) to clear the denominators:\[x^3 - 2x^2 - 4x + 3 = Ax^3 + Bx^2 + Cx + D\]This equation will allow us to find the values of \(A\), \(B\), \(C\), and \(D\).

Step 3: Equate Coefficients

Equate the coefficients from both sides of the equation:1. For \(x^3\): \(1 = A\)2. For \(x^2\): \(-2 = B\)3. For \(x\): \(-4 = C\)4. Constant term: \(3 = D\)From this, we directly obtain the values:\(A = 1\), \(B = -2\), \(C = -4\), and \(D = 3\).

Step 4: Write the Final Decomposition

Substitute the values of the constants \(A\), \(B\), \(C\), and \(D\) back into the partial fraction form:\[\frac{x^3 - 2x^2 - 4x + 3}{x^4} = \frac{1}{x} + \frac{-2}{x^2} + \frac{-4}{x^3} + \frac{3}{x^4}\]This is the partial fraction decomposition of the given rational function.

Key Concepts

Rational Functions

Rational functions are key components when dealing with partial fractions. They are mathematical expressions in the form of \( \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\). These functions behave quite differently from regular polynomials because they can have discontinuities or 'holes' where the denominator equals zero.
In the context of partial fraction decomposition, rational functions are broken into simpler fractions that are easier to handle in integrations or further calculations. This breaking down helps automate solutions to differential equations, evaluate integral calculations, and solve systems of equations.
Understanding rational functions' anatomy is crucial:

• The numerator \(P(x)\) dictates the overall degree of the function.
• The denominator \(Q(x)\) is crucial for identifying the 'shape' of the function, and influences both its domain and behavior at certain points.

To work effectively with them, it's important to analyze their degree and anticipate requirements such as polynomial division when the degree of the numerator exceeds the denominator.

Repeated Linear Factors

In partial fraction decomposition, rational functions with repeated linear factors in the denominator require special attention. Repeated factors mean that a factor appears more than once, often represented as \((x - r)^n\). For instance, if your denominator includes \(x^4\), it indicates repeated factors of \(x\) up to the fourth power.
When you decompose such a rational function, each power of the factor must be accounted for separately. This is why in our example, the partial fraction decomposition includes terms for each power of \(x\) from 1 to 4: \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} \).

• The presence of repeated linear factors causes the setup of partial fractions to have multiple terms.
• Each exponent must be represented with a unique constant to solve for, hence increasing the number of unknowns in the equation.

Understanding how to handle these factors is crucial in ensuring all potential solutions are accounted for during decomposition.

Polynomial Long Division

Polynomial long division is a technique used when dealing with rational functions where the numerator's degree is equal to or greater than the denominator's. This process is akin to numerical long division but with polynomials.
The goal is to simplify the rational function to a proper form where the numerator's degree is less than the denominator's. If the given rational function doesn't require this due to being already simplified or inverted, the solution phase moves directly to decomposition.

• This method helps break down complex expressions into manageable parts.
• After division, the resulting quotient is often a polynomial (or simpler rational function), which will form part of our final answer.

Remember, while polynomial long division can seem complex, it's merely a systematic approach to dividing polynomial expressions and is essential when simplifying expressions with higher-degree numerators. This step ensures that decomposition only focuses on what remains post-division.