1. What is Partial Fraction Decomposition?

Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions that are easier to work with, especially for integration or inverse Laplace transforms.

2. How Does the Calculator Work?

The calculator decomposes rational functions of the form:

Where:

• \( P(x) \) — Numerator polynomial
• \( Q(x) \) — Denominator polynomial (factored)
• \( A_i \) — Coefficients to be determined
• \( r_i \) — Roots of the denominator

Explanation: The method involves factoring the denominator, setting up equations for the coefficients, and solving the system of equations.

3. Importance of Partial Fractions

Details: Partial fractions are essential in calculus for integration, in differential equations for solving with Laplace transforms, and in control theory for system analysis.

4. Using the Calculator

Tips: Enter the numerator and denominator polynomials in standard form. For denominators, use factored form when possible (e.g., (x+1)(x-2)).

5. Frequently Asked Questions (FAQ)

Q1: What types of denominators can be processed?
A: This calculator handles denominators with linear factors. Repeated roots and irreducible quadratics require more advanced methods.

Q2: Why is partial fraction decomposition useful?
A: It simplifies complex rational expressions, making them easier to integrate, differentiate, or analyze.

Q3: What if my denominator can't be factored?
A: The method requires factorable denominators. You may need to use numerical methods or more advanced techniques for irreducible polynomials.

Q4: Can this handle improper fractions?
A: No, the numerator degree must be less than the denominator degree. Perform polynomial division first if needed.

Q5: Are there limitations to this calculator?
A: It currently handles distinct linear factors. Future versions may include repeated roots and quadratic factors.