Algebraic Expression
An algebraic expression is a mathematical phrase that combines numbers, variables, and operations but contains no equals sign.
Definition
An algebraic expression is a math phrase built from numbers, variables, and operations such as addition, subtraction, multiplication, division, or exponentiation. Unlike an equation, it has no equals sign, so it cannot be "solved," only evaluated or simplified. Formally, an algebraic expression is a well-formed formula over a set of variables and constants under a signature of operations; in abstract algebra, expressions are elements of a free algebra. Rational expressions (quotients of polynomials) extend the concept to include division, with domain restrictions wherever the denominator would equal zero.
Example
$3x + 7$ is an algebraic expression, and so is $2x^2 + 5x - 3$, which has three terms; evaluating the latter at $x = 2$ gives $2(4) + 10 - 3 = 15$. The rational expression $(x^2-4)/(x-2)$ simplifies to $x + 2$ for all $x \neq 2$; the restriction $x \neq 2$ must be stated explicitly because the original expression is undefined there.
Key Insight
An expression is like a recipe that gives you a result once you know the ingredient amounts, but it never makes a claim that two things are equal; expressions are the building blocks of equations and inequalities. Free algebras formalize the idea that an expression's structure is independent of any particular numeric interpretation, allowing symbolic manipulation as a mathematical object in its own right.