Rational Expression
A rational expression is a fraction where both the numerator and denominator are polynomials, defined for all values that do not make the denominator zero.
Formula
\frac{P(x)}{Q(x)}, \; Q(x) \neq 0
Definition
A rational expression is a fraction whose numerator and denominator are both polynomials, $P(x)/Q(x)$ with $Q(x)$ not the zero polynomial; like ordinary fractions, the denominator can never equal zero, so its domain excludes any value of $x$ where $Q(x) = 0$. To simplify, factor both numerator and denominator and cancel common factors, but note that a restriction from a canceled factor still applies to the domain even after the factor is gone. Formally, rational expressions form the field of fractions of the polynomial ring $F[x]$, denoted $F(x)$: elements are equivalence classes of pairs $(P, Q)$ with $Q \neq 0$, where $(P, Q) \sim (P', Q')$ if $PQ' = P'Q$, and this construction is the function field of the projective line $\mathbb{P}^1$ over $F$, the starting point for studying maps between curves via rational functions in algebraic geometry.
Example
$(x + 3)/(x - 2)$ is undefined when $x = 2$; at $x = 5$, its value is $8/3$. $(x^2 - 4)/(x - 2) = (x+2)(x-2)/(x-2) = x + 2$ for $x \neq 2$, and that restriction still applies even after the factor cancels. In $F(x)$, $(x^2-1)/(x-1)$ and $(x+1)/1$ are equal as elements since $(x^2-1) \cdot 1 = (x+1)(x-1)$, though as functions they differ at $x = 1$.
Key Insight
Rational expressions work just like numeric fractions: you can simplify, multiply, divide, add, and subtract them, always watching for division by zero. Canceling a factor removes the "hole" from the formula but never the domain restriction it created, so always state restrictions before simplifying, a subtlety that matters because $\mathbb{Q}[x]$ being an integral domain is exactly what makes the field-of-fractions construction possible.