Rational Expression
AlgebraA 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 where the top and bottom are polynomials. Like regular fractions, the bottom (denominator) cannot equal zero.
Example
$(x + 3)/(x - 2)$ is a rational expression. It is undefined when $x = 2$ (denominator $= 0$). When $x = 5$, the value is $(5+3)/(5-2) = 8/3$.
Key Insight
Rational expressions work just like fractions with numbers: you can simplify, multiply, divide, add, and subtract them, always watching out for division by zero.
Definition
A rational expression is the quotient $P(x)/Q(x)$ of two polynomials, where $Q(x)$ is not the zero polynomial. The domain excludes values of $x$ where $Q(x) = 0$. To simplify, factor both numerator and denominator and cancel common factors. Restrictions must be noted before canceling, because a canceled factor still restricts the domain.
Example
$(x^2 - 4)/(x - 2) = (x+2)(x-2)/(x-2) = x + 2$ for $x$ not equal to $2$. The restriction $x$ not equal to $2$ applies even though it cancelled.
Key Insight
Canceling a factor from a rational expression changes the formula (removes the hole in the graph) but the domain restriction remains. Always state restrictions before simplifying.
Definition
Rational expressions form the field of fractions of the polynomial ring $F[x]$, denoted $F(x)$. This field is the smallest field containing $F[x]$ as a subring. Elements of $F(x)$ are equivalence classes of pairs $(P, Q)$ with $Q \neq 0$, where $(P, Q) \sim (P', Q')$ iff $PQ' = P'Q$. Operations mirror those of fractions over integers ($\mathbb{Q}[x]$ is an integral domain, enabling field-of-fractions construction).
Example
In $F(x)$, $(x^2 - 1)/(x - 1)$ and $(x + 1)/1$ are equal as elements of $F(x)$ since $(x^2-1) \cdot 1 = (x+1) \cdot (x-1)$. But as functions, they differ at $x = 1$.
Key Insight
The field of rational functions $F(x)$ is the function field of the projective line $\mathbb{P}^1$ over $F$. Studying maps between curves via rational functions is the starting point of algebraic geometry over function fields.