Algebraic Expression
An algebraic expression is a combination of variables, constants, and arithmetic operations (addition, subtraction, multiplication, division, exponentiation) that represents a mathematical quantity without an equality sign. Unlike equations, expressions do not assert that two things are equal — they simply describe a value that depends on the variables present. Examples include 3x + 5, 2a² − 4b, and (x + y)/2.
Worked Example
Problem
Evaluate the expression 4x² − 3x + 7 when x = 2.
Solution
Step 1: Substitute x = 2 into the expression. 4(2)² − 3(2) + 7 Step 2: Evaluate the exponent. 4(4) − 3(2) + 7 Step 3: Perform multiplications. 16 − 6 + 7 Step 4: Add and subtract left to right. 10 + 7 = 17
Answer
17
Types of Algebraic Expressions
| Type | Number of Terms | Example | Description |
|---|---|---|---|
| Monomial | 1 | 5x² | Single term with coefficient and variable(s) |
| Binomial | 2 | 3x − 4 | Sum or difference of two monomials |
| Trinomial | 3 | x² + 2x − 1 | Sum of exactly three monomials |
| Polynomial | 2 or more | 4x³ − x² + 6x − 9 | General term for multi-term expressions |
| Rational expression | Fraction form | (x + 1)/(x − 2) | Ratio of two polynomials |
Interactive Tools
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Related Terms
Variable (algebra)
A variable is a symbol, typically a letter such as x, y, or n, that represents an unknown or changing quantity in a mathematical expression or equation. Variables allow mathematicians to write general rules and relationships that apply to many specific cases at once. In algebra, manipulating variables to solve for unknowns or express patterns is the central skill.
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
Equation (algebra)
An algebraic equation is a mathematical statement asserting that two expressions are equal, connected by an equals sign (=). Solving an equation means finding the value(s) of the variable(s) that make the statement true, called the solution or root. Equations are fundamental to all branches of mathematics and science, providing a precise language for describing quantitative relationships.
From Arabic "al-jabr" (the reunion of broken parts), introduced by Persian mathematician Muhammad ibn Musa al-Khwarizmi in his 9th-century treatise "Kitab al-mukhtasar fi hisab al-jabr wal-muqabala". The word "expression" comes from Latin "expressio" meaning a pressing out or representation.