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 calculus and differential equations.

2. How Does the Calculator Work?

The calculator solves equations of the form:

Where:

• \( p \) — Coefficient of x in numerator
• \( q \) — Constant term in numerator
• \( a, b \) — Roots of the denominator
• \( A, B \) — Coefficients to be found

Method: The calculator uses the Heaviside cover-up method to determine the coefficients A and B by solving the resulting system of equations.

3. Importance of Partial Fractions

Applications: Partial fraction decomposition is essential for integration of rational functions, solving differential equations, and performing inverse Laplace transforms in engineering and physics.

4. Using the Calculator

Instructions: Enter the coefficients from your rational expression. The numerator should be in the form px + q, and the denominator should be factorable into (x + a)(x + b).

5. Frequently Asked Questions (FAQ)

Q1: What if my denominator has repeated roots?
A: This calculator handles distinct linear factors. For repeated roots, additional terms with higher powers are needed in the decomposition.

Q2: Can this handle quadratic factors?
A: No, this calculator is designed for linear factors only. Quadratic factors require a different approach.

Q3: Why is partial fraction decomposition useful?
A: It simplifies complex rational expressions, making them easier to integrate or transform in calculus and engineering applications.

Q4: What if my numerator degree is higher than denominator?
A: You must first perform polynomial long division before applying partial fraction decomposition.

Q5: Can this be used for improper fractions?
A: Only proper fractions (where numerator degree is less than denominator degree) can be directly decomposed.