Partial Fraction Coefficient Calculator

Partial Fraction Decomposition:

\[ \frac{ax + b}{(x - c)(x - d)} = \frac{A}{x-c} + \frac{B}{x-d} \]

Coefficient A:

Coefficient B:

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1. What is Partial Fraction Decomposition?

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions that are easier to work with, especially in calculus and differential equations.

2. How Does the Calculator Work?

The calculator uses the following formulas:

\[ A = \frac{a c + b}{c - d} \] \[ B = \frac{a d + b}{d - c} \]

Where:

\( a \) — Coefficient of x in the numerator
\( b \) — Constant term in the numerator
\( c \) — First root in the denominator
\( d \) — Second root in the denominator
\( A \) — Coefficient for the first partial fraction
\( B \) — Coefficient for the second partial fraction

Explanation: The formulas are derived by equating coefficients and solving the resulting system of equations.

3. Importance of Partial Fractions

Details: Partial fractions are essential for integration, solving differential equations, and performing inverse Laplace transforms in engineering and physics.

4. Using the Calculator

Tips: Enter all coefficients and roots. The roots (c and d) must be distinct for this decomposition to work. The calculator will compute the coefficients A and B for the partial fractions.

5. Frequently Asked Questions (FAQ)

Q1: What if the denominator has repeated roots?
A: This calculator handles distinct roots only. For repeated roots, the decomposition form is different.

Q2: Can this handle higher degree polynomials?
A: This calculator is designed for linear numerators and quadratic denominators with distinct roots.

Q3: What if the roots are complex?
A: The formulas work for complex numbers, but this calculator displays real results only.

Q4: Why is partial fraction decomposition useful?
A: It simplifies integration problems and helps solve differential equations more easily.

Q5: What if the denominator can't be factored?
A: This calculator requires factorable denominators with distinct roots.