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 uses the partial fraction decomposition formula:

Where:

• \( a \) — Coefficient of x in numerator
• \( b \) — Constant term in numerator
• \( c \) — Coefficient of x in denominator
• \( d \) — Constant term in denominator
• \( A \) — First partial fraction coefficient
• \( B \) — Second partial fraction coefficient

Explanation: The equation is decomposed into simpler fractions that can be more easily integrated or analyzed.

3. Importance of Partial Fractions

Details: Partial fractions are essential in calculus for integration, in Laplace transforms for solving differential equations, and in simplifying complex rational expressions.

4. Using the Calculator

Tips: Enter the coefficients (a, c) and constants (b, d) from your rational expression. All values must be valid numbers (c and d cannot be zero).

5. Frequently Asked Questions (FAQ)

Q1: What types of expressions can this calculator handle?
A: This calculator handles simple linear factors of the form (ax + b)/(cx + d). More complex denominators require different approaches.

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 has repeated roots?
A: This calculator doesn't handle repeated roots. Those require additional terms in the decomposition.

Q4: Can this be used for improper fractions?
A: No, the numerator degree must be less than the denominator degree for partial fractions.

Q5: What are common applications of partial fractions?
A: Integration in calculus, solving differential equations with Laplace transforms, and simplifying transfer functions in engineering.