Terms & Expressions
The building blocks of every algebraic statement — understanding these deeply sets up everything ahead.
Bringing it Together
In the first two lessons you learned about variables (letters representing unknowns) and coefficients (numbers attached to variables) and constants (numbers standing alone). Now we name the structures they form: terms and expressions.
These two words are used constantly throughout algebra — and throughout mathematics at large. Using them precisely signals fluency. By the end of this lesson you’ll be reading and describing any algebraic expression with confidence.
What is a Term?A term is a single algebraic unit. It is either a number, a variable, or a product of numbers and variables. The critical property of a term: it contains no addition or subtraction. Terms are the individual pieces that expressions are built from.
An algebraic expression is a combination of one or more terms, connected by addition or subtraction. It represents a value — one that depends on whatever the variable(s) equal.
An equation has an equals sign and asserts two things are equal: 3x + 5 = 14
We solve equations. We simplify or evaluate expressions. These are different tasks.
Expressions are classified by how many terms they contain. These names come up frequently in later lessons, especially when studying polynomials (Lesson 69 onwards).
Let’s take one expression and break it down completely into its parts:
Three terms separated by + and −
Notice how the minus sign before the 4x belongs to that term. The coefficient of x is −4, not 4. The sign between terms always travels with the term that follows it.
In 5x − 3y + 8, the three terms are +5x, −3y, and +8. The first term is positive by default.
Two terms are called like terms if they have identical variable parts — the same variable(s) raised to the same power(s). Their coefficients can differ; only the variable part must match.
We’ll use this idea extensively in Lesson 11 (Collecting Like Terms), but recognising like terms starts here.
| Pair | Like Terms? | Reason |
|---|---|---|
| 3x and 7x | ✓ Yes | Both have variable part x (to the power 1) |
| 5y² and −2y² | ✓ Yes | Both have variable part y² |
| 4x and 4x² | ✗ No | x ≠ x² — the powers differ |
| 6ab and −ab | ✓ Yes | Both have variable part ab |
| 3x and 3y | ✗ No | x ≠ y — the variables differ |
| 9 and −4 | ✓ Yes | Both are constants — constants are always like terms with each other |
You can add 3 apples + 5 apples = 8 apples. They’re the same type of fruit — like terms.
But 3 apples + 5 oranges cannot be combined into a single fruit type — unlike terms. The best you can say is “3 apples and 5 oranges.”
Algebra works exactly the same way. 3x + 5y cannot be simplified further — x and y are different “fruits.”
Every term has a degree — the sum of the exponents (powers) of all variables in that term. This number describes the term’s “complexity” and becomes important in Lesson 69 (Polynomials).
| Term | Exponents | Degree |
|---|---|---|
| 7 | No variable (exponent = 0) | 0 — constant term |
| 5x | x¹ | 1 — linear term |
| 3x² | x² | 2 — quadratic term |
| 4x³ | x³ | 3 — cubic term |
| 2x²y | x² × y¹ → exponents: 2 + 1 | 3 |
| ab²c | a¹ × b² × c¹ → 1 + 2 + 1 | 4 |
For each expression: count the terms, classify the expression (monomial/binomial/trinomial), and list each term with its sign.
From the expression 5x + 3y − 2x + 4y² + 7 − 1, group the like terms together.
State the degree of each term.
(a) 6x (b) x + 4 (c) 3a² − 2a + 1 (d) 5mn − 3m + 2n − 8
(b) 2 terms → binomial
(c) 3 terms → trinomial
(d) 4 terms → polynomial
The sign before each term belongs to that term. The first term is positive by default.
(a) 4x and 4y (b) 3a² and 7a² (c) 5xy and −2xy (d) x² and x³
(b) Yes — both have variable part a²
(c) Yes — both have variable part xy
(d) No — different powers (x² ≠ x³)
(a) 5x⁴ (b) 3x²y (c) −7 (d) ab³c²
The expression would just be: 4x − 3 (no equals sign, no claim about its value).
y terms: 5y and 4y
Constants: 3 and −2
We haven’t combined them yet — that comes in Lesson 11. Right now, just identifying the groups is the skill.
· 3x²: coefficient 3, variable x², degree 2
· −7x: coefficient −7, variable x, degree 1
· 4: constant, degree 0
- A term is a single unit: a number, variable, or their product — no + or − inside
- An expression is one or more terms joined by + or −
- An expression has no equals sign — it represents a value but doesn’t claim what it equals
- Expressions are named by term count: monomial, binomial, trinomial, polynomial
- Like terms share identical variable parts (same variables, same powers)
- The sign before a term belongs to that term — it’s part of the coefficient
- The degree of a term is the sum of all its variable exponents