Partial Fraction Decomposition

Calculus
process

Also known as: partial fractions, fraction decomposition

Grade 9-12

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Breaking a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator. Partial fractions turn the integration of any rational function into a collection of standard integrals (logarithms and arctangents).

This concept is covered in depth in our complete partial fraction decomposition tutorial, with worked examples, practice problems, and common mistakes.

Definition

Breaking a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator.

💡 Intuition

Just as \frac{7}{12} can be split into \frac{1}{3} + \frac{1}{4}, a complex fraction like \frac{5x-1}{(x+1)(x-2)} can be split into \frac{A}{x+1} + \frac{B}{x-2}. The simpler pieces are each easy to integrate.

🎯 Core Idea

Every proper rational function can be decomposed into partial fractions. The form depends on the factors of the denominator: linear factors give \frac{A}{x-a}, repeated factors give \frac{A}{(x-a)^k}, and irreducible quadratics give \frac{Ax+B}{x^2+bx+c}.

Example

\int \frac{5x-1}{(x+1)(x-2)}\,dx
Decompose: \frac{5x-1}{(x+1)(x-2)} = \frac{2}{x+1} + \frac{3}{x-2}.
= 2\ln|x+1| + 3\ln|x-2| + C

Formula

\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b} Repeated: \frac{P(x)}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}. Irreducible quadratic: \frac{Ax+B}{x^2+bx+c}.

Notation

\frac{P(x)}{Q(x)} = proper rational function (deg P < deg Q). A, B, C, ... are constants to be determined.

🌟 Why It Matters

Partial fractions turn the integration of any rational function into a collection of standard integrals (logarithms and arctangents). It's also used in Laplace transforms, differential equations, and signal processing.

💭 Hint When Stuck

Factor the denominator completely first, then write the template with unknowns A, B, C and use strategic x-values to solve for them quickly.

Formal View

If \deg P < \deg Q and Q(x) = (x-a_1)^{m_1} \cdots (x-a_k)^{m_k}(x^2+b_1x+c_1)^{n_1} \cdots, then \frac{P(x)}{Q(x)} = \sum_{i=1}^{k} \sum_{j=1}^{m_i} \frac{A_{ij}}{(x-a_i)^j} + \sum_{i=1} \sum_{j=1}^{n_i} \frac{B_{ij}x + C_{ij}}{(x^2+b_ix+c_i)^j} for unique constants A_{ij}, B_{ij}, C_{ij}.

🚧 Common Stuck Point

Make sure the fraction is proper (degree of numerator < degree of denominator) before decomposing. If it's improper, do polynomial long division first.

⚠️ Common Mistakes

  • Forgetting to do long division when the degree of the numerator is greater than or equal to the degree of the denominator: \frac{x^3}{x^2-1} must be divided first.
  • Using the wrong form for repeated factors: (x-1)^2 in the denominator requires \frac{A}{x-1} + \frac{B}{(x-1)^2}, NOT just \frac{A}{(x-1)^2}.
  • Using the wrong form for irreducible quadratic factors: x^2 + 1 requires \frac{Ax + B}{x^2+1}, NOT \frac{A}{x^2+1}—you need a linear numerator.

Frequently Asked Questions

What is Partial Fraction Decomposition in Math?

Breaking a rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator.

What is the Partial Fraction Decomposition formula?

\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b} Repeated: \frac{P(x)}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}. Irreducible quadratic: \frac{Ax+B}{x^2+bx+c}.

When do you use Partial Fraction Decomposition?

Factor the denominator completely first, then write the template with unknowns A, B, C and use strategic x-values to solve for them quickly.

Prerequisites

integrallong division

How Partial Fraction Decomposition Connects to Other Ideas

To understand partial fraction decomposition, you should first be comfortable with integral and long division. Once you have a solid grasp of partial fraction decomposition, you can move on to improper integrals.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Partial Fraction Decomposition: Step-by-Step Guide →
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