1. What Is Partial Fraction Decomposition?

Partial fraction decomposition is a mathematical technique that breaks down a rational function (ratio of two polynomials) into a sum of simpler fractions. This is particularly useful in calculus for integration and in control theory for Laplace transforms.

2. How The Algorithm Works

The decomposition process follows these systematic steps:

Algorithm Steps:

1. Factor the denominator Q(x) completely
2. For each distinct linear factor (x - r), include term A/(x - r)
3. For repeated linear factors (x - r)^n, include terms A₁/(x - r) + A₂/(x - r)² + ... + Aₙ/(x - r)ⁿ
4. For irreducible quadratic factors, include terms of form (Bx + C)/(ax² + bx + c)
5. Set up equation and solve for unknown coefficients

3. Applications of Partial Fractions

Key Applications: Integration of rational functions, inverse Laplace transforms, solving differential equations, and simplifying complex algebraic expressions in engineering and physics problems.

4. Using This Calculator

Instructions: Enter the numerator and denominator polynomials in standard mathematical notation. Use 'x' as the variable, and include parentheses for factors. The calculator will show step-by-step decomposition.

5. Frequently Asked Questions (FAQ)

Q1: What types of factors can be handled?
A: The algorithm handles distinct linear factors, repeated linear factors, and irreducible quadratic factors.

Q2: How are repeated factors treated?
A: For a factor (x - r)^n, we include n terms: A₁/(x - r) + A₂/(x - r)² + ... + Aₙ/(x - r)ⁿ.

Q3: What if the numerator degree is greater than denominator?
A: First perform polynomial division, then decompose the proper rational function remainder.

Q4: Can complex roots be handled?
A: Complex roots come in conjugate pairs, creating irreducible quadratic factors in the decomposition.

Q5: What notation should I use for input?
A: Use standard polynomial notation: 3x^2 + 2x - 1, and factored form: (x-1)(x+2)(x^2+1).