Partial Fraction Decomposition Calculator
Enter a polynomial like 3x + 5 or x^2 - 2
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What is Partial Fraction Decomposition?
Partial fraction decomposition is a technique used to break down complex rational expressions (fractions with polynomials) into simpler fractions. This process is particularly useful in calculus for integrating rational functions and in solving differential equations.
When to Use Partial Fraction Decomposition
Partial fraction decomposition is typically used when:
- You need to integrate a complex rational function
- You're working with rational expressions in differential equations
- You want to simplify complex fractions for easier manipulation
- You're finding inverse Laplace transforms in engineering and physics
How Partial Fraction Decomposition Works
The process involves several key steps:
- Ensure the numerator's degree is less than the denominator's (perform long division if needed)
- Factor the denominator completely
- Write out the partial fraction decomposition form based on the factors
- Solve for the unknown coefficients using various methods (equating coefficients, substituting values, etc.)
Types of Factors and Their Decomposition
Linear Factors: (ax + b)
For each linear factor (ax + b) that appears n times in the denominator, the decomposition includes terms of the form:
Quadratic Factors: (ax² + bx + c)
For irreducible quadratic factors (ax² + bx + c) that appear n times, the decomposition includes:
Example of Partial Fraction Decomposition
Example: Decompose (3x + 5)/((x + 1)(x - 2))
Step 1: Write the partial fraction form
Step 2: Multiply both sides by (x + 1)(x - 2)
Step 3: Find the coefficients by substituting values
When x = -1: 3(-1) + 5 = A(-1 - 2) + B(0) = -3A
Therefore: 2 = -3A, so A = -2/3
When x = 2: 3(2) + 5 = A(0) + B(2 + 1) = 3B
Therefore: 11 = 3B, so B = 11/3
Step 4: Write the final decomposition