1. What is Partial Fractions Decomposition?

Partial fractions decomposition is a technique in algebra that breaks down complex rational expressions into simpler fractions that are easier to work with, especially for integration in calculus.

2. How Does the Calculator Work?

The calculator decomposes rational expressions using the general formula:

Where:

• \( P(x) \) — Numerator polynomial
• \( Q(x) \) — Denominator polynomial (factored)
• \( A_i, B_j, C_j \) — Constants to be determined
• \( r_i \) — Real roots of denominator
• \( x^2 + p_jx + q_j \) — Irreducible quadratic factors

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

3. Importance of Partial Fractions

Details: Partial fractions are essential for integrating rational functions in calculus, solving differential equations, and simplifying complex algebraic expressions.

4. Using the Calculator

Tips: Enter the numerator and denominator polynomials. The denominator should be in factored form (e.g., (x+1)(x-2) or x^2-4).

5. Frequently Asked Questions (FAQ)

Q1: What types of denominators can be processed?
A: The calculator handles denominators with linear factors, repeated linear factors, and irreducible quadratic factors.

Q2: What if my denominator can't be factored?
A: The denominator must be factorable into linear and/or irreducible quadratic factors for partial fractions decomposition to work.

Q3: How are repeated roots handled?
A: For a factor (x-a)^n, the decomposition includes terms with denominators (x-a), (x-a)^2, ..., up to (x-a)^n.

Q4: Can this handle complex roots?
A: Complex roots come in conjugate pairs and are combined into irreducible quadratic factors in the real number system.

Q5: What's the main application of partial fractions?
A: The primary use is in calculus for integrating rational functions, but it's also useful in Laplace transforms and other areas.