1. What Is A Partial Fraction?

A partial fraction is a way of breaking down a complex rational expression into simpler fractions that can be more easily integrated or analyzed. It's particularly useful in calculus for integration and in control theory for Laplace transforms.

2. How Partial Fraction Decomposition Works

The basic form for two distinct linear factors:

Where:

• \( A \) and \( B \) — Coefficients to be determined
• \( a \) and \( b \) — Roots of the denominator
• \( x \) — Variable in the expression

Process: The decomposition involves finding constants A and B such that the sum of the partial fractions equals the original fraction.

3. Applications of Partial Fractions

Key Uses:

• Integration of rational functions in calculus
• Solving differential equations
• Inverse Laplace transforms in engineering
• Simplifying complex algebraic expressions

4. Using the Calculator

Instructions: Enter the coefficients (A, B), roots (a, b), and variable name (default is x). The calculator will display the decomposed form.

5. Frequently Asked Questions (FAQ)

Q1: What if the denominator has repeated roots?
A: For repeated roots (x-a)², the decomposition includes terms with increasing powers: A/(x-a) + B/(x-a)².

Q2: Can this handle quadratic factors?
A: Yes, irreducible quadratics in the denominator require terms of the form (Ax+B)/(x²+cx+d).

Q3: How are the coefficients determined?
A: Typically by equating coefficients or substituting strategic values for x.

Q4: Why is partial fraction decomposition useful?
A: It transforms complex fractions into simpler terms that are easier to integrate or transform.

Q5: What's the difference between proper and improper fractions?
A: Proper fractions have numerator degree less than denominator. Improper fractions require polynomial division first.