Partial Fraction Decomposition Calculator

Decompose rational functions into partial fractions with step-by-step solutions

Numerator

Use * for multiplication, ^ for exponents (e.g., x^2 for x²)

Denominator

Enter polynomial or factored form

Variable

Examples:

Numerator

Use * for ×, ^ for power

Denominator

Variable

About 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.

This method is particularly useful in calculus for integration, solving differential equations, and in engineering for inverse Laplace transforms and signal processing.

By decomposing a rational function into partial fractions, you can integrate complex expressions term by term, making otherwise difficult problems much more manageable.

Key Benefits

• Integration: Simplifies integration of rational functions

• Laplace Transforms: Essential for inverse transforms

• Differential Equations: Helps solve complex equations

• Signal Processing: Useful in control theory and filters

Types of Partial Fractions

Linear Factors

Form:

For denominator (ax + b)(cx + d):

Decompose as A/(ax + b) + B/(cx + d)

Example:

1/(x² - 1) = 1/((x-1)(x+1))

= 1/2(x-1) - 1/2(x+1)

Repeated Factors

Form:

For denominator (ax + b)ⁿ:

Decompose as A₁/(ax + b) + A₂/(ax + b)² + ... + Aₙ/(ax + b)ⁿ

Example:

1/(x-1)² = A/(x-1) + B/(x-1)²

How to Perform Partial Fraction Decomposition

Step-by-Step Process

Check if proper fraction: Ensure degree of numerator < degree of denominator. If not, use polynomial division first.
Factor the denominator: Factor completely into linear and irreducible quadratic factors.
Set up the partial fraction form: Write the sum of partial fractions based on the factors.
Clear denominators: Multiply both sides by the common denominator.
Solve for coefficients: Use substitution or equate coefficients to find A, B, C, etc.
Write final answer: Express the original fraction as sum of partial fractions.

Real-World Applications

Calculus Integration

Essential for integrating complex rational functions by breaking them into simpler logarithmic and arctangent forms.

Control Systems

Used in control theory to find inverse Laplace transforms and analyze system stability and response.

Signal Processing

Helpful in filter design and analyzing frequency responses in electrical engineering applications.

Common Examples

Example 1: Distinct Linear Factors

Problem: (x + 7)/(x² + 3x + 2)

Factor: (x + 7)/((x + 1)(x + 2))

Solution: 6/(x + 1) - 5/(x + 2)

Example 2: Simple Fraction

Problem: 1/(x² - 1)

Factor: 1/((x - 1)(x + 1))

Solution: 1/(2(x - 1)) - 1/(2(x + 1))

Tips for Success

Best Practices:

• Always factor the denominator completely first
• Check if the fraction is proper before decomposing
• Use substitution method for faster coefficient solving
• Verify your answer by combining the partial fractions

Common Mistakes to Avoid:

• Forgetting to include all repeated factor terms
• Incorrectly factoring the denominator
• Not checking if fraction is proper first
• Making algebraic errors when solving for coefficients

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