Partial Fractions: Decomposition Method and Examples

Partial fraction decomposition is the reverse of adding algebraic fractions: it splits a single complicated fraction like (5x + 6)/((x + 1)(x + 2)) into a sum of simpler fractions, here 1/(x + 1) + 4/(x + 2). Each simpler piece is easy to work with on its own.

This matters because the technique is a workhorse of calculus and engineering. Splitting a rational function into partial fractions is the standard first step for integrating it, and it is essential for solving differential equations with Laplace transforms in control engineering.

Key Concepts

• Proper fraction: a rational expression where the numerator has lower degree than the denominator. Only proper fractions can be decomposed directly.
• Distinct linear factors: a denominator like (x + 1)(x + 2) with two different linear factors. Each factor gets its own partial fraction with a constant numerator.
• Cover-up method: a shortcut for finding each constant: cover up one factor, substitute its root into what remains.
• Repeated and quadratic factors: denominators like (x + 1) squared or (x squared + 1) need extra terms, but the basic idea is the same.

How It Works

1. Check the fraction is proper. If not, divide the numerator by the denominator first and decompose only the remainder part.
2. Factorise the denominator completely into linear factors.
3. Write one partial fraction per factor: (5x + 6)/((x + 1)(x + 2)) = A/(x + 1) + B/(x + 2).
4. Multiply through by the denominator and substitute the roots x = -1 and x = -2 to find A and B (the cover-up method).
5. Recombine your answer over a common denominator to verify it matches the original fraction.

Worked Example

Decompose (5x + 6)/((x + 1)(x + 2)).

1. The fraction is proper (degree 1 over degree 2), and the denominator is already factored.
2. Set (5x + 6)/((x + 1)(x + 2)) = A/(x + 1) + B/(x + 2).
3. Multiply through: 5x + 6 = A(x + 2) + B(x + 1).
4. Put x = -1: 5(-1) + 6 = 1 = A(1), so A = 1.
5. Put x = -2: 5(-2) + 6 = -4 = B(-1), so B = 4.
6. Answer: 1/(x + 1) + 4/(x + 2).

Verify with x = 0: original = 6/(1 times 2) = 3; decomposed = 1/1 + 4/2 = 1 + 2 = 3. They match.

Common Misconceptions

• You can decompose any fraction straight away. Wrong. If the numerator's degree is not lower than the denominator's, you must do polynomial division first.
• Each factor always gets just one constant term. Wrong for repeated factors: (x + 1) squared needs both A/(x + 1) and B/((x + 1) squared).
• The cover-up substitution works with any x value. It only gives a constant directly when you substitute the root that makes all other factors vanish; any other value gives an equation to solve instead.

Real-World Applications

• In calculus, partial fractions turn hard integrals of rational functions into a sum of easy logarithmic integrals.
• In control engineering, Laplace transforms of system responses are inverted to time-domain signals via partial fractions.
• In signal processing, breaking a transfer function into simpler parallel blocks uses exactly this decomposition.

To recap: make the fraction proper, factorise the denominator, assign one constant per linear factor, and find the constants with the cover-up method. Always recombine to check your decomposition reproduces the original fraction.