Rational expressions
Also known as: algebraic fractions
A rational expression is a fraction whose numerator and denominator are polynomials, such as (x² − 9)/(x + 3). They follow the same rules as numeric fractions, with the added restriction that the denominator cannot equal zero.
A rational expression is a ratio of two polynomials — an algebraic fraction like (x² − 9)/(x + 3) or 5/(2x − 1). The name comes from "ratio," just as rational numbers are ratios of integers. Because division by zero is undefined, any value of the variable that makes the denominator zero is excluded from the expression's domain: (x + 2)/(x − 4) is undefined at x = 4.
Simplifying a rational expression means factoring the numerator and denominator and canceling common factors. For example, (x² − 9)/(x + 3) factors to (x + 3)(x − 3)/(x + 3), which simplifies to x − 3 (for x ≠ −3). The classic mistake is canceling terms instead of factors — you cannot cancel the x's in (x + 5)/x, because the x in the numerator is a term of a sum, not a factor of the whole numerator.
Operations on rational expressions mirror fraction arithmetic. To multiply, factor everything and cancel across the fractions; to divide, multiply by the reciprocal; to add or subtract, rewrite each expression over a common denominator first, then combine the numerators. Keeping track of excluded values through each step is part of a complete answer.
Standardized tests lean on these skills. The ACT tests simplifying expressions and combining algebraic fractions directly, while math competitions like the AMC 10 and 12 embed rational expressions inside polynomial and equation-solving problems. Fluent factoring is the prerequisite — most rational expression errors are really factoring errors.
Key takeaways
- A rational expression is a polynomial divided by a polynomial.
- Values that make the denominator zero are excluded from the domain.
- Simplify by factoring and canceling common factors — never cancel individual terms.
- Add and subtract over a common denominator; divide by multiplying by the reciprocal.
- Strong factoring skills are the foundation for every rational expression problem.