Partial Fraction Decomposition
Partial fraction decomposition is a method used to rewrite rational expressions—fractions where both numerator and denominator are polynomials—into a sum of simpler fractions. This technique is essential in calculus, Laplace transforms, and system analysis in engineering.
Form of a Rational Expression:
Where $P(x)$ and $Q(x)$ are polynomials, and the degree of the numerator is less than the degree of the denominator.
Why It Works:
If the denominator can be factored, the rational expression can be decomposed into a sum of simpler terms whose denominators match those factors. This makes integration and algebraic manipulation easier.
Case 1: Distinct Linear Factors
Case 2: Repeated Linear Factors
Case 3: Irreducible Quadratic Factors
Steps in Decomposition:
- Ensure the degree of the numerator is less than that of the denominator. If not, perform polynomial division.
- Factor the denominator completely.
- Set up partial fractions based on the type of factors (linear, repeated, or quadratic).
- Multiply through by the common denominator to eliminate fractions.
- Solve for constants by substitution or comparing coefficients.
This decomposition method transforms complex expressions into manageable parts for integration, inverse transforms, or system modeling in engineering contexts.
