Partial Fraction Method Calculator With Solution

Partial Fraction Decomposition:

\[ \frac{x}{x^2 - 4} = \frac{A}{x-2} + \frac{B}{x+2} \]

Coefficient A:

Coefficient B:

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1. What is Partial Fraction Decomposition?

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions that are easier to work with, especially in integration and Laplace transforms.

2. How Does the Calculator Work?

The calculator decomposes rational expressions of the form:

\[ \frac{P(x)}{Q(x)} = \frac{A}{(x-a)} + \frac{B}{(x-b)} + \cdots \]

Where:

\( P(x) \) — Numerator polynomial
\( Q(x) \) — Denominator polynomial (factorizable)
\( A, B, \ldots \) — Coefficients to be determined

3. Step-by-Step Solution

Example Solution:

Enter values and click Calculate to see the step-by-step solution.

4. Using the Calculator

Tips: Enter the numerator and denominator polynomials. The denominator must be factorizable into linear factors for this method to work.

5. Frequently Asked Questions (FAQ)

Q1: What types of denominators can be used?
A: This calculator handles denominators that can be factored into distinct linear factors. Repeated roots and irreducible quadratics require different approaches.

Q2: Why use partial fractions?
A: They simplify complex rational expressions, making them easier to integrate or transform.

Q3: What if the numerator degree is higher?
A: Polynomial long division must be performed first until the numerator degree is less than the denominator.

Q4: Can this handle complex roots?
A: Not in this basic version. Complex roots require pairing conjugate factors.

Q5: What's the most common application?
A: Integration of rational functions in calculus is the primary application.