Partial Fractions
Partial fraction decomposition is the process of expressing a rational expression as a sum of simpler fractions whose denominators are the factors of the original denominator. This technique reverses the process of adding fractions and is essential for integrating rational functions in calculus, solving differential equations, and performing inverse Laplace transforms in engineering. Each simpler fraction is called a partial fraction.
Key Formula
P(x)/[(ax+b)(cx+d)] = A/(ax+b) + B/(cx+d)
LaTeX: \frac{P(x)}{(ax+b)(cx+d)} = \frac{A}{ax+b} + \frac{B}{cx+d}
| Symbol | Meaning | Unit |
|---|---|---|
| P(x) | numerator polynomial (degree < degree of denominator) | dimensionless |
| A, B | unknown constants to be determined | dimensionless |
| ax+b, cx+d | distinct linear factors of the denominator | dimensionless |
Worked Example
Problem
Decompose (3x + 5) / [(x + 1)(x + 2)] into partial fractions.
Solution
Step 1: Write (3x + 5) / [(x + 1)(x + 2)] = A/(x + 1) + B/(x + 2). Step 2: Multiply both sides by (x + 1)(x + 2): 3x + 5 = A(x + 2) + B(x + 1). Step 3: Substitute x = −1: 3(−1) + 5 = A(1) + 0 → 2 = A, so A = 2. Step 4: Substitute x = −2: 3(−2) + 5 = 0 + B(−1) → −1 = −B, so B = 1. Step 5: Result: 2/(x + 1) + 1/(x + 2).
Answer
(3x + 5) / [(x+1)(x+2)] = 2/(x+1) + 1/(x+2)
Partial Fraction Forms by Denominator Factor Type
| Factor Type | Example Factor | Partial Fraction Form | Notes |
|---|---|---|---|
| Distinct linear | (x − a) | A/(x − a) | One constant per factor |
| Repeated linear | (x − a)² | A/(x−a) + B/(x−a)² | One term per power |
| Irreducible quadratic | (x²+bx+c) | (Ax+B)/(x²+bx+c) | Linear numerator needed |
| Repeated quadratic | (x²+1)² | (Ax+B)/(x²+1)+(Cx+D)/(x²+1)² | Two linear numerators |
| Improper fraction | degree P ≥ degree Q | Polynomial + proper fractions | Divide first |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Rational Expression
A rational expression is a fraction in which both the numerator and denominator are polynomials, and the denominator is not zero. It represents the ratio of two polynomial functions and is defined for all values of the variable except those that make the denominator equal to zero. Rational expressions are used in modeling real-world phenomena, simplifying algebraic fractions, and solving equations in calculus and engineering.
Rational Function
A rational function is a function that can be expressed as the ratio of two polynomials, f(x) = P(x)/Q(x), where Q(x) ≠ 0. Its domain excludes all values of x that make the denominator zero, and its graph exhibits vertical asymptotes at these excluded values and may have horizontal or oblique asymptotes determined by the degrees of P and Q. Rational functions model phenomena such as average cost, concentration over time in pharmacokinetics, and electrical impedance in circuits.
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
The term "partial fraction" derives from the idea of splitting a fraction into its component "parts." The technique was developed in the 17th and 18th centuries, with contributions from Johann Bernoulli and Gottfried Wilhelm Leibniz as part of the development of integral calculus.