1Cademy - Rational Expression

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Polynomial

Definition

Rational Expression

A rational expression is an expression written in the form $\frac{p(x)}{q(x)}$ , where $p$ and $q$ are polynomials and $q \neq 0$ . Just as a rational number is defined as the ratio of two integers with a nonzero denominator, a rational expression extends this idea by replacing integers with polynomials — it is the ratio of two polynomials whose denominator polynomial is not equal to zero. Because division by zero is undefined, the denominator must never be zero.

Examples of rational expressions include $-\frac{13}{42}$ , $\frac{7y}{8z}$ , $\frac{5x + 2}{x^2 - 7}$ , and $\frac{4x^2 + 3x - 1}{2x - 8}$ . Notice that $-\frac{13}{42}$ is simply a numerical fraction, yet it qualifies as a rational expression because a constant is a polynomial of degree zero — so the ratio of two constants is a ratio of two polynomials, provided the denominator is not zero. The same operations performed on numerical fractions — simplifying, adding, subtracting, multiplying, and dividing — are also performed on rational expressions.

Updated 2026-05-01

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