1. What is Partial Fraction Decomposition?

Partial fraction decomposition breaks down a rational function (ratio of polynomials) into simpler fractions that can be more easily integrated, transformed, or analyzed.

2. How Does the Calculator Work?

The calculator decomposes a rational function:

Where:

• \( P(x) \) — Numerator polynomial
• \( Q(x) \) — Denominator polynomial
• \( r_i \) — Roots of the denominator
• \( m_i \) — Multiplicity of each root
• \( A_{ij} \) — Coefficients to be determined

3. Importance of Partial Fractions

Applications: Essential for integration in calculus, solving differential equations, and Laplace transforms in engineering.

4. Using the Calculator with Desmos

Tips: Enter numerator and denominator polynomials. The calculator will show the decomposition and you can visualize both the original function and its partial fractions in the embedded Desmos graphing calculator.

5. Frequently Asked Questions (FAQ)

Q1: What polynomial formats are accepted?
A: Standard form like "x^2+3x+2" or factored form like "(x+1)(x+2)". Use 'x' as the variable.

Q2: How are repeated roots handled?
A: Each root appears in the decomposition according to its multiplicity in the denominator.

Q3: Can I plot the results in Desmos?
A: Yes, the embedded Desmos calculator allows you to visualize both the original function and its partial fraction components.

Q4: What about improper fractions?
A: The calculator automatically handles cases where the degree of numerator ≥ denominator by performing polynomial division first.

Q5: Are complex roots supported?
A: Currently only real roots are shown in the decomposition, but complex roots can be visualized in Desmos.