Partial Fractions Calculator

Partial Fractions Formula:

\[ \frac{1}{x(x+1)} = \frac{A}{x} + \frac{B}{x+1} \]

Partial Fractions Result:

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1. What is Partial Fractions Decomposition?

Definition: Partial fractions is a technique for breaking down complex rational expressions into simpler fractions that are easier to work with.

Purpose: Used in calculus for integration, in differential equations, and in Laplace transforms to simplify complex fractions.

2. How Does Partial Fractions Work?

The general approach:

\[ \frac{P(x)}{Q(x)} = \frac{A}{x-a} + \frac{B}{x-b} + \cdots \]

Where:

\( P(x) \) — Numerator polynomial
\( Q(x) \) — Denominator polynomial (factored)
\( A, B \) — Constants to be determined

Steps:

Factor the denominator completely
Set up partial fractions with unknown constants
Multiply through by the denominator
Solve for the constants

3. Example Calculation Steps

4. When to Use Partial Fractions

Applications:

Integrating rational functions
Solving differential equations
Simplifying complex algebraic expressions
Inverse Laplace transforms

5. Frequently Asked Questions (FAQ)

Q1: What if the denominator can't be factored?
A: Partial fractions can only be applied to factorable denominators. You may need numerical methods instead.

Q2: How do I handle repeated roots?
A: For (x-a)^n, include terms with A/(x-a), B/(x-a)^2, ..., up to N/(x-a)^n.

Q3: What if the numerator degree is higher than denominator?
A: First perform polynomial long division, then apply partial fractions to the remainder.

Q4: Are there different methods to find the constants?
A: Yes, besides substitution, you can use the cover-up method or equate coefficients.

Q5: Can this handle complex roots?
A: Yes, but they must come in conjugate pairs for real coefficients.