Adding Rational Expressions

Intuition and core definition

A rational expression is a quotient of polynomials. Addition requires a common denominator because numerators count units of that denominator. Denominators may be multiplied by missing factors, but excluded values from every original denominator remain excluded even if later factors cancel.

Notation, language, and conditions

For $A/C+B/C=(A+B)/C$ with $C\ne0$. The least common denominator (LCD) contains each irreducible factor to the greatest power appearing. Domain restrictions are values making any original denominator zero; they belong to the expression’s definition, not merely the simplified form.

Why this idea matters

Adding rational expressions requires a common denominator that preserves each original domain restriction, just as fraction addition requires common-sized parts.

A dependable method

1. Factor every denominator completely and state excluded values.
2. Build the LCD using each necessary factor at maximum multiplicity.
3. Multiply each numerator and denominator by its missing factor.
4. Add the entire adjusted numerators, using parentheses.
5. Factor and simplify the result while retaining original restrictions.

Worked example

Common mistakes