Adding & Subtracting Rational Expressions

Rational expressions are fractions with polynomials. Adding or subtracting them works just like numeric fractions: you need a common denominator first.

Find the least common denominator, rewrite each fraction over it, then add or subtract the numerators. Keep the denominator and simplify.

Find the common denominator

The common denominator is the product of the distinct denominators (or their least common multiple if they share factors). Multiply each fraction by the missing factor.

For $\dfrac{4}{x} + \dfrac{2}{x + 3}$ , the common denominator is $x(x + 3)$ . Rewrite both fractions over it.

Combine numerators

Once denominators match, add or subtract the numerators and keep the common denominator. Distribute carefully, especially with subtraction.

Simplify the result and note any values that make the original denominators zero — those are excluded from the domain.

Worked examples

Example 1: adding two fractions

Add $\dfrac{4}{x} + \dfrac{2}{x + 3}$ .

Common denominator

x(x + 3)

Rewrite each numerator

\dfrac{4(x+3) + 2x}{x(x+3)}

Combine

\dfrac{6x + 12}{x(x+3)}

Answer: $\dfrac{6x + 12}{x(x + 3)}$

Example 2: like denominators

Add $\dfrac{2}{x} + \dfrac{5}{x}$ .

Same denominator, add numerators

\dfrac{2 + 5}{x}

Simplify

\dfrac{7}{x}

Answer: $\dfrac{7}{x}$

Example 3: subtracting with unlike denominators

Subtract $\dfrac{3}{x - 1} - \dfrac{2}{x}$ .

Common denominator

x(x - 1)

Rewrite each numerator

\dfrac{3x - 2(x - 1)}{x(x - 1)}

Distribute the minus across both terms

\dfrac{3x - 2x + 2}{x(x - 1)}

Combine

\dfrac{x + 2}{x(x - 1)}

Answer: $\dfrac{x + 2}{x(x - 1)}$

Try one yourself

Common questions

What is the common denominator here?

The least common multiple of the denominators. If they share no factors, it is just their product, like $x(x+3)$ .

What is the biggest mistake?

Forgetting to distribute a subtraction across the whole second numerator, which flips only some signs by accident.

Do I need to state excluded values?

Yes — any x that makes an original denominator zero is excluded from the domain, even after simplifying.

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Adding & Subtracting Rational Expressions

Find the common denominator

Combine numerators

Worked examples

Example 1: adding two fractions

Example 2: like denominators

Example 3: subtracting with unlike denominators

Try one yourself

Common questions

Keep learning

Multiplying & Dividing Rational Expressions

Graphing Rational Functions

Solving Rational Equations & Inequalities