Method Of Partial Fractions Calculator Symbolab

Partial Fractions Decomposition:

\[ \frac{x^2 + 1}{x(x+2)^2} = \frac{A}{x} + \frac{B}{x+2} + \frac{C}{(x+2)^2} \]

Coefficient A:

Coefficient B:

Coefficient C:

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1. What is Partial Fractions Decomposition?

Partial fractions 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 solves equations of the form:

\[ \frac{P(x)}{Q(x)} = \frac{A}{x-a} + \frac{B}{(x-b)^2} + \cdots \]

Where:

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

Method: The calculator uses the Symbolab method of equating coefficients and solving the resulting system of linear equations.

3. Importance of Partial Fractions

Applications: Essential for integration of rational functions, solving differential equations, and Laplace transforms in engineering and physics.

4. Using the Calculator

Instructions: Enter the numerator and denominator polynomials. The denominator should be in factored form for best results.

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: How are the coefficients determined?
A: By multiplying through by the denominator and solving the resulting system of equations for the unknown coefficients.

Q3: Can it handle complex roots?
A: This version handles real roots only. Complex roots require a different approach.

Q4: What's the maximum degree polynomial supported?
A: The calculator can handle polynomials up to 4th degree in the current implementation.

Q5: Why use partial fractions in integration?
A: It transforms complex rational expressions into simpler terms that can be integrated using basic logarithmic and arctangent forms.