🍰 Algebra Solver

Partial Fraction Decomposition Calculator

Decompose rational expressions into simpler partial fractions step-by-step. Solve partial fraction decomposition for algebra and integration.

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How to Use & Educational Guide

In-depth guide for Partial Fraction Decomposition Calculator

Educational Overview

What Is Partial Fraction Decomposition?

Partial fraction decomposition is a method for rewriting a rational expression as a sum of simpler fractions. It is commonly used when a polynomial fraction has a factored denominator and is especially useful for algebra, calculus, and integration. For example, a complicated fraction such as $\frac{3x+5}{(x+1)(x+2)}$ can be rewritten as simpler fractions that are easier to work with.

Partial Fraction Decomposition Formula

When the denominator contains two distinct linear factors, the basic partial fraction form is:

$$\frac{Ax+B}{(x+p)(x+q)} = \frac{C}{x+p} + \frac{D}{x+q}$$

The constants $C$ and $D$ are determined by clearing the denominator and solving the resulting equations.

Partial Fraction Decomposition Rules

The form used depends on how the denominator is factored. The main rules are:

Distinct linear factors: Use one constant numerator over each factor, such as $\frac{A}{x-a} + \frac{B}{x-b}$.
Repeated linear factors: Include a separate fraction for every power, such as $\frac{A}{x-a} + \frac{B}{(x-a)^2}$.
Irreducible quadratic factors: Use a linear numerator, such as $\frac{Ax+B}{x^2+px+q}$.
Improper rational expressions: If the numerator degree is greater than or equal to the denominator degree, perform polynomial long division first.

How to Do Partial Fraction Decomposition?

Step 1 (Check the Fraction): Make sure the degree of the numerator is less than the degree of the denominator. If not, use polynomial long division first.
Step 2 (Factor the Denominator): Completely factor the denominator into linear or irreducible quadratic factors.
Step 3 (Write the Partial Fractions): Set up the correct decomposition form based on the denominator factors.
Step 4 (Clear the Denominators): Multiply the entire equation by the common denominator to remove all fractions.
Step 5 (Find the Unknown Constants): Substitute convenient values of $x$ or compare coefficients to determine the unknown constants.
Step 6 (Write the Final Decomposition): Substitute the constants into the partial fractions and simplify the result.

Partial Fraction Decomposition Examples

Example 1: Decompose $\frac{3x+5}{x^2+3x+2}$

• Factor the denominator: $x^2+3x+2=(x+1)(x+2)$.

• Set up the decomposition: $\frac{3x+5}{(x+1)(x+2)}=\frac{A}{x+1}+\frac{B}{x+2}$.

• Clear denominators: $3x+5=A(x+2)+B(x+1)$.

• Set $x=-1$: $2=A$, so $\mathbf{A=2}$.

• Set $x=-2$: $-1=-B$, so $\mathbf{B=1}$.

👉 Result: $\frac{2}{x+1}+\frac{1}{x+2}$

Example 2: Decompose $\frac{1}{x^2-4}$

• Factor: $x^2-4=(x-2)(x+2)$.

• Set up: $\frac{1}{(x-2)(x+2)}=\frac{A}{x-2}+\frac{B}{x+2}$.

• Clear denominators: $1=A(x+2)+B(x-2)$.

• Set $x=2$: $1=4A$, so $\mathbf{A=\frac14}$.

• Set $x=-2$: $1=-4B$, so $\mathbf{B=-\frac14}$.

👉 Result: $\frac{1}{4(x-2)}-\frac{1}{4(x+2)}$

Example 3: Repeated Factor

• Expression: $\frac{1}{(x-1)^2(x+2)}$.

• Correct form: $\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x+2}$.

• Key rule: A repeated factor such as $(x-1)^2$ requires a separate partial fraction for each power.

👉 Decomposition Form: $\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x+2}$

When to Use Partial Fraction Decomposition?

∫ Calculus Integration

Partial fractions can turn a rational function into simpler terms that are easier to integrate, especially fractions involving linear or quadratic factors.

🎓 Algebra & Polynomial Work

The method helps rewrite rational expressions into simpler components for algebraic manipulation, equation solving, and further calculations.

Common Mistakes to Avoid

Skipping Factorization: Always factor the denominator completely before choosing the partial fraction form.
Using the Wrong Numerator: A linear or irreducible quadratic factor requires a linear numerator such as $Ax+B$, not just a constant.
Forgetting Repeated Powers: A factor like $(x-a)^3$ requires terms for all powers: $\frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{(x-a)^3}$.
Ignoring Improper Fractions: If the numerator degree is equal to or greater than the denominator degree, perform polynomial division before decomposition.

Mathematical accuracy verified by AptCalc Engine

Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

What is partial fraction decomposition?

Partial fraction decomposition is a method of rewriting a rational expression as a sum of simpler fractions. It is commonly used in algebra and calculus integration.

How do you do partial fraction decomposition?

Factor the denominator, write the correct partial fraction form, clear the denominators, solve for the unknown constants, and substitute them into the decomposition.

When should you use partial fraction decomposition?

It is especially useful for simplifying rational expressions and integrating rational functions in calculus.