1.49 Rational Expression Definitions

Rational Expression Definitions establishes the vocabulary for algebraic fractions in which polynomials appear in the numerator and denominator, describing the permissible input values over which such an expression is defined, the specific values that must be excluded from that domain, and the criteria for judging when two differently written rational expressions represent the same underlying value.

Rational Expression

A rational expression is a fraction in which both the numerator and the denominator are polynomials, such as (x + 3)/(x² − 4). Rational expressions extend the concept of a numerical fraction into algebra, behaving according to the same rules governing addition, subtraction, multiplication, and division of fractions, but with polynomials taking the place of simple numbers in the numerator and denominator.

Rational Domain

The rational domain of a rational expression is the set of every real number that may be validly substituted for the variable without causing the denominator to equal zero. Since division by zero is undefined, the rational domain always excludes any value that would make the denominator vanish, even while the numerator may be evaluated at that same value without any difficulty.

Excluded Value

An excluded value is a specific number that is not part of a rational expression's domain because substituting it for the variable would make the denominator equal zero. For the rational expression (x + 3)/(x² − 4), the excluded values are 2 and −2, since substituting either value causes the denominator x² − 4 to equal zero, making the expression undefined at those two specific points.

Rational Equivalence

Rational equivalence is the property that two rational expressions, despite being written differently, represent exactly the same value for every input shared by both of their domains, typically established by multiplying or dividing the numerator and denominator of one expression by the same nonzero polynomial factor. The rational expressions (x + 3)/(x² − 4) and 1/(x − 2), obtained by canceling a shared factor of (x + 3) from a differently written but equivalent numerator and denominator, satisfy rational equivalence wherever both expressions remain defined, even though their written forms differ.

Together, these definitions establish the rational expression as an algebraic fraction of polynomials, with its rational domain bounded by whatever excluded values would otherwise cause division by zero, and rational equivalence providing the standard for recognizing when two differently written rational expressions describe the same underlying relationship.