Algebraic Expressions — Terms, Coefficient, Constant, Like/Unlike Terms

Question

What are the parts of an algebraic expression — terms, coefficients, constants, and how do we identify like and unlike terms?

Solution — Step by Step

An algebraic expression is a combination of variables, constants, and operations. Terms are the parts separated by $+$ or $-$ signs.

Example: $3x^2 + 5xy - 7x + 2$

This has 4 terms: $3x^2$ , $5xy$ , $-7x$ , and $2$ .

Notice that $-7x$ carries its negative sign — the sign belongs to the term.

Coefficient = the numerical part multiplying the variable(s) in a term.

In $3x^2$ : coefficient is 3
In $-7x$ : coefficient is $-7$
In $xy$ : coefficient is 1 (hidden, but there)

Constant = a term with no variable at all.

In $3x^2 + 5xy - 7x + 2$ : the constant term is 2

Like terms have the SAME variable part (same variables raised to the same powers). Only the coefficients may differ.

$3x^2$ and $-5x^2$ are like terms (both have $x^2$ )
$7xy$ and $2xy$ are like terms (both have $xy$ )
$3x^2$ and $3x$ are unlike terms ( $x^2 \neq x$ )
$4xy$ and $4x^2y$ are unlike terms ( $xy \neq x^2y$ )

We can only add or subtract like terms. Unlike terms stay as they are.

graph TD
    A[Parts of an Algebraic Expression] --> B[Terms: separated by + or -]
    B --> C[Variable terms: contain letters]
    B --> D[Constant terms: pure numbers]
    C --> E[Coefficient: numerical multiplier]
    C --> F[Variable part: letters with powers]
    F --> G{Same variable part?}
    G -->|Yes| H[Like terms: can be combined]
    G -->|No| I[Unlike terms: cannot be combined]

Why This Works

Algebra is a language. Variables are placeholders for unknown numbers, and coefficients tell us “how many” of that unknown we have. Like terms can be combined because they represent the same type of quantity — just as 3 apples + 5 apples = 8 apples, $3x + 5x = 8x$ .

We cannot combine unlike terms for the same reason we cannot add apples and oranges — $3x + 5y$ stays as $3x + 5y$ .

Alternative Method

A quick test for like terms: cover the coefficients and compare what remains. If the leftover parts are identical, the terms are like.

$7x^2y$ and $-3x^2y$ : cover 7 and $-3$ , both leave $x^2y$ — like terms
$5x^2$ and $5x^3$ : cover 5 and 5, one leaves $x^2$ , other leaves $x^3$ — unlike terms

Common Mistake

Students often say $x$ and $x^2$ are like terms because “both have $x$ .” They are NOT like terms. The power matters. $x = x^1$ and $x^2$ have different exponents, so they are unlike terms and cannot be added. Always check that both the variable AND its power match.