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Partial Fraction Decomposition Calculator

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

Enter Rational Function

Numerator Polynomial

Numerator: 5x^2 + 7

Denominator Polynomial

Denominator: x^3 - 2x^2 - 2x - 3

Rational Function

5x^2 + 7
x^3 - 2x^2 - 2x - 3

Partial Fraction Decomposition

4/x - 3 + x - 1/x² + x + 1
4
x - 3
x - 1
x² + x + 1

Step-by-Step Solution

1

Original rational function

f(x) = (5x^2 + 7) / (x^3 - 2x^2 - 2x - 3)
2

Factor the denominator

x^3 - 2x^2 - 2x - 3 = (x - 3)(x² + x + 1)
3

Set up partial fraction form

(5x^2 + 7)/((x - 3)(x² + x + 1)) = A/(x - 3) + (Bx + C)/(x² + x + 1)
4

Clear denominators and solve for coefficients

5x² + 7 = A(x² + x + 1) + (Bx + C)(x - 3)
5

Expand and collect like terms

5x² + 7 = (A + B)x² + (A - 3B + C)x + (A - 3C)
6

Compare coefficients

A + B = 5, A - 3B + C = 0, A - 3C = 7
7

Solve the system of equations

A = 4, B = 1, C = -1
8

Final partial fraction decomposition

(5x^2 + 7)/((x - 3)(x² + x + 1)) = 4/(x - 3) + (x - 1)/(x² + x + 1)

Quick Examples

Decomposition Rules

1

Linear Factors

For (ax + b): A/(ax + b)

2

Repeated Linear

For (ax + b)²: A/(ax + b) + B/(ax + b)²

3

Quadratic

For (ax² + bx + c): (Ax + B)/(ax² + bx + c)

4

Improper

If deg(numerator) ≥ deg(denominator), divide first

Solution Methods

✓

Clear denominators by multiplying both sides

✓

Expand and collect like terms

✓

Compare coefficients of each power

✓

Solve the resulting system of equations

✓

Substitute back into partial fraction form

Understanding Partial Fraction Decomposition

What is Partial Fraction Decomposition?

Partial fraction decomposition is the process of breaking down a complex rational function (a quotient of polynomials) into a sum of simpler rational functions. This technique is particularly useful in calculus for integration, differential equations, and Laplace transforms.

Why Use Partial Fractions?

  • •Simplifies integration of rational functions
  • •Useful in solving differential equations
  • •Essential for inverse Laplace transforms
  • •Helps in analyzing systems and control theory

General Process

Step 1: Check Degree

If degree of numerator ≥ degree of denominator, perform polynomial long division first.

Step 2: Factor Denominator

Factor the denominator into irreducible linear and quadratic factors.

Step 3: Set Up Partial Fractions

Write the decomposition with unknown coefficients based on factor types.

Step 4: Solve for Coefficients

Clear denominators and solve the resulting system of equations.

Common Examples

Linear Factors

5x + 2 / (x - 1)(x + 3)
↓
A/(x - 1) + B/(x + 3)

Repeated Linear Factor

3x - 5 / (x - 2)²(x + 1)
↓
A/(x - 2) + B/(x - 2)² + C/(x + 1)

Quadratic Factor

x² + 1 / (x - 1)(x² + x + 1)
↓
A/(x - 1) + (Bx + C)/(x² + x + 1)

Mixed Factors

2x³ + x + 1 / x²(x² + 4)
↓
A/x + B/x² + (Cx + D)/(x² + 4)

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