Rational Expression

A rational expression is the algebraic analogue of a rational number — it has a polynomial numerator and a polynomial denominator: $\frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$.

Simplifying means factoring numerator and denominator and cancelling common factors. Example: $\frac{x^2 - 1}{x + 1} = \frac{(x-1)(x+1)}{x+1} = x - 1$ (for $x \neq -1$).

Domain restrictions matter: any value making the original denominator zero must be excluded, even if it cancels in simplification. Above, $x = -1$ is excluded from the domain even though the simplified form $x - 1$ would accept it.

Operations: addition / subtraction (find common denominator), multiplication (multiply across, then simplify), division (multiply by reciprocal). Rational expressions are the foundation of partial fraction decomposition used in integration.