Algebraic Expressions: Types, Operations & Examples - Complete Guide

What Is an Algebraic Expression?

An algebraic expression is a mathematical phrase that combines variables (like x, y), constants (like 3, -7), and operations (addition, subtraction, multiplication, division). Unlike an equation, an expression does not have an equals sign.

Example: 3x² + 5x - 7 is an expression. 3x² + 5x - 7 = 0 is an equation.

Key Idea: Every expression is built from terms - parts separated by + or - signs. In 3x² + 5x - 7, the three terms are 3x², 5x, and -7.

Parts of an Expression

Part	What It Is	Example
Variable	A symbol for an unknown number	x, y, z
Coefficient	The number multiplied by a variable	In 5x, the coefficient is 5
Constant	A number with no variable	-7, 3, 12
Term	Each part separated by + or -	3x², 5x, -7

Types of Algebraic Expressions

Expressions are named by how many terms they have:

Monomial - one term: 5x, -3y², 8
Binomial - two terms: x + 3, 2a - 5b
Trinomial - three terms: x² + 2x + 1
Polynomial - any number of terms (general term): 4x³ - 2x² + x - 6

Operations on Algebraic Expressions

1. Adding and Subtracting

Only combine like terms - terms with the same variable and same exponent.

3x + 5x = 8x (same variable x)
4y² - y² = 3y² (same variable and exponent)
3x + 5y = 3x + 5y (cannot combine - different variables)

2. Multiplying

Use the distributive property: a(b + c) = ab + ac

2(x + 3) = 2x + 6
3(2x - 4) = 6x - 12
(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 (FOIL method)

3. Dividing

Divide each term in the numerator by the denominator, or factor and cancel common factors.

(6x + 9) / 3 = 2x + 3
(x² - 4) / (x + 2) = x - 2

Try It: Simplify 2(3x - 1) + 4(x + 5).

Step 1 - Distribute: 6x - 2 + 4x + 20
Step 2 - Combine like terms: (6x + 4x) + (-2 + 20) = 10x + 18

FOIL Method for Multiplying Binomials

When multiplying two binomials, use FOIL: First, Outer, Inner, Last.

Example: (x + 4)(x - 2)

First: x × x = x²
Outer: x × (-2) = -2x
Inner: 4 × x = 4x
Last: 4 × (-2) = -8

Combine: x² - 2x + 4x - 8 = x² + 2x - 8

Common Mistakes

Combining unlike terms: 3x + 2y does NOT equal 5xy. You can only combine like terms.
Forgetting to distribute: 2(x + 3) is NOT 2x + 3. It's 2x + 6.
Sign errors: When subtracting, distribute the negative: -(x - 4) = -x + 4, not -x - 4.

Remember: An expression is simplified when no more like terms can be combined and no more distributive operations remain. Always check your answer by substituting a test value.

Try our Solve for X Calculator or Polynomial Calculator for more practice.