1. What is the Partial Fraction Method?

The Partial Fraction Method is a technique in algebra and calculus that decomposes a rational function into simpler fractions. This is particularly useful for integration, Laplace transforms, and solving differential equations.

2. How Does the Calculator Work?

The calculator decomposes the rational function:

Where:

• \( N(s) \) — Numerator polynomial
• \( D(s) \) — Denominator polynomial (factored)
• \( A, B, \ldots, Z \) — Constants to be determined
• \( p_1, p_2, \ldots, p_n \) — Roots of denominator

Explanation: The method solves for unknown coefficients by either the Heaviside cover-up method (for distinct roots) or algebraic method (for repeated/complex roots).

3. Importance of Partial Fractions

Details: Partial fraction decomposition is essential for integrating rational functions, solving linear differential equations, and performing inverse Laplace transforms in engineering and physics.

4. Using the Calculator

Tips: Enter the numerator and denominator polynomials (e.g., "s^2+3s+2" or "(s+1)(s+2)"). Select the method - Heaviside for distinct roots, Algebraic for repeated roots.

5. Frequently Asked Questions (FAQ)

Q1: What polynomial formats are accepted?
A: Both expanded (e.g., s^2+3s+2) and factored (e.g., (s+1)(s+2)) forms are accepted.

Q2: How are repeated roots handled?
A: The algebraic method must be used for repeated roots, creating terms with increasing powers in the denominator.

Q3: Can complex roots be processed?
A: Yes, though the result may include complex numbers or quadratic terms in the denominator.

Q4: What's the Heaviside cover-up method?
A: A shortcut technique for finding coefficients when roots are distinct, by "covering up" factors and evaluating at roots.

Q5: How accurate is this calculator?
A: The calculator performs symbolic computation, so results are mathematically exact when inputs are properly formatted.