Fractions in Algebra — Definition, Formula & Examples

Fractions in algebra are fractions where the numerator, denominator, or both contain variables or algebraic expressions. They follow the same rules as numeric fractions — you can add, subtract, multiply, divide, and simplify them — but you work with expressions like

x + 2

3a

instead of plain numbers.

An algebraic fraction (also called a rational expression) is a quotient of two algebraic expressions of the form $\frac{P}{Q}$ , where $P$ and $Q$ are polynomials and $Q \neq 0$ . Operations on algebraic fractions require the same procedures as arithmetic fractions — common denominators for addition and subtraction, and factoring for simplification — extended to polynomial expressions.

How It Works

To simplify an algebraic fraction, factor both the numerator and denominator, then cancel common factors. To add or subtract algebraic fractions, find a common denominator (usually the least common denominator), rewrite each fraction, then combine the numerators. Multiplication works by multiplying numerators together and denominators together, then simplifying. Division means multiplying by the reciprocal of the second fraction. Always state restrictions: any value of the variable that makes a denominator zero is excluded from the domain.

Worked Example

Problem: Simplify:

\frac{x^2 - 9}{x^2 + 5x + 6}

Factor the numerator: Recognize a difference of squares.

x^2 - 9 = (x - 3)(x + 3)

Factor the denominator: Find two numbers that multiply to 6 and add to 5: those are 2 and 3.

x^2 + 5x + 6 = (x + 2)(x + 3)

Cancel common factors: The factor

(x + 3)

appears in both, so cancel it (noting

x \neq -3

\frac{(x-3)\cancel{(x+3)}}{(x+2)\cancel{(x+3)}} = \frac{x - 3}{x + 2}

Answer:

\frac{x - 3}{x + 2}

, where

x \neq -3

and

x \neq -2

Why It Matters

Algebraic fractions appear throughout high school algebra and precalculus — in solving rational equations, simplifying complex expressions, and working with rates or proportions in physics. Mastering them is essential preparation for calculus, where techniques like partial fraction decomposition rely directly on these skills.

Common Mistakes

Mistake: Canceling terms instead of factors — for example, canceling the

x^2

\frac{x^2 + 1}{x^2 + 3}

to get

\frac{1}{3}

Correction: You can only cancel factors that multiply the entire numerator and denominator. Since

x^2

is added (not multiplied) within each expression, it cannot be canceled. Factor first, then cancel.