1. What Is Partial Fraction Decomposition?

Partial fraction decomposition is a technique in algebra and calculus that breaks down a rational function into simpler fractions that are easier to integrate, differentiate, or analyze. It's particularly useful for solving integrals of rational functions.

2. How Does The Calculator Work?

The calculator uses algorithmic methods to decompose rational functions:

Where:

• \( P(x) \) — Numerator polynomial
• \( Q(x) \) — Denominator polynomial (factored form)
• \( r_1, r_2, \ldots, r_n \) — Roots of the denominator
• \( A_1, A_2, \ldots, A_n \) — Constants to be determined

Process: The algorithm factors the denominator, sets up equations for the constants, and solves the system to find the partial fractions.

3. Importance Of Partial Fractions

Applications: Essential for integration techniques, solving differential equations, Laplace transforms, and simplifying complex rational expressions in engineering and physics problems.

4. Using The Calculator

Instructions: Enter the numerator and denominator polynomials in standard algebraic notation. Use proper syntax (e.g., "3x^2 + 2x - 1" for numerator and "(x-1)(x+2)" for denominator).

5. Frequently Asked Questions (FAQ)

Q1: What types of denominators are supported?
A: The calculator handles linear factors, repeated linear factors, and irreducible quadratic factors in the denominator.

Q2: How accurate is the decomposition?
A: The algorithm provides exact symbolic decomposition when possible, following standard algebraic procedures.

Q3: Can it handle complex roots?
A: Yes, the calculator can decompose functions with complex roots, though results are typically presented with real coefficients when possible.

Q4: What's the maximum degree supported?
A: The calculator can handle polynomials up to 10th degree, though computation time may increase with complexity.

Q5: How are repeated factors handled?
A: Repeated factors are decomposed using the appropriate form with multiple terms for each repetition.