Partial Fraction Decomposition
Partial fraction decomposition breaks a complex rational expression into simpler fractions, essential for integration and Laplace transforms.
The Formula
Partial fraction decomposition rewrites a complicated fraction as a sum of simpler fractions. This technique is essential for integrating rational functions and for inverse Laplace transforms.
The method depends on the type of factors in the denominator: distinct linear factors, repeated linear factors, or irreducible quadratic factors each have their own decomposition pattern.
Decomposition Rules
| Denominator Factor | Decomposition Form |
|---|---|
| Distinct linear: (x − a) | A / (x − a) |
| Repeated linear: (x − a)² | A / (x − a) + B / (x − a)² |
| Irreducible quadratic: (x² + bx + c) | (Ax + B) / (x² + bx + c) |
| Repeated quadratic: (x² + bx + c)² | (Ax + B) / (x² + bx + c) + (Cx + D) / (x² + bx + c)² |
Example 1
Decompose (3x + 5) / ((x + 1)(x − 2)) into partial fractions.
Set up: (3x + 5) / ((x + 1)(x − 2)) = A/(x + 1) + B/(x − 2)
Multiply both sides by (x + 1)(x − 2): 3x + 5 = A(x − 2) + B(x + 1)
Set x = 2: 3(2) + 5 = A(0) + B(3) → 11 = 3B → B = 11/3
Set x = −1: 3(−1) + 5 = A(−3) + B(0) → 2 = −3A → A = −2/3
(3x + 5) / ((x + 1)(x − 2)) = −2/(3(x + 1)) + 11/(3(x − 2))
Example 2
Decompose (5x + 3) / (x²(x + 1)) into partial fractions.
x² is a repeated linear factor, so: A/x + B/x² + C/(x + 1)
Multiply through: 5x + 3 = Ax(x + 1) + B(x + 1) + Cx²
Set x = 0: 3 = B(1) → B = 3
Set x = −1: −5 + 3 = C(1) → C = −2
Expand and compare x² coefficients: 0 = A + C → A = 2
(5x + 3) / (x²(x + 1)) = 2/x + 3/x² − 2/(x + 1)
When to Use It
Partial fractions appear throughout calculus, differential equations, and engineering.
- Integrating rational functions (turns hard integrals into simple log and arctan forms)
- Inverse Laplace transforms in control theory and signal processing
- Solving linear differential equations with constant coefficients
- Simplifying algebraic expressions in circuit analysis
- Z-transform inversions in digital signal processing
Important: before decomposing, always check that the degree of the numerator is less than the degree of the denominator. If not, perform polynomial long division first.