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A term is a single number, variable, or product of numbers and variables separated from other terms by top-level addition or subtraction.
Terms are the structural units of expressions. Correct term boundaries make distribution, collection of like terms, polynomial degree, and factoring possible.
Intuition and core definition
A term is a single number, variable, or product of numbers and variables separated from other terms by top-level addition or subtraction. In $4x^2-3xy+7$, the terms are $4x^2$, $-3xy$, and $7$. The sign immediately before a term belongs to that term.
Notation, language, and conditions
A term may contain factors joined by multiplication and powers. Parentheses can keep a sum inside one factor: in $5(x+2)$, the expression has one product before expansion, while $5x+10$ has two terms. Like terms have identical variable parts with identical exponents, so only their coefficients differ.
Why this idea matters
A term is a signed additive component, so identifying term boundaries prevents illegal cancellation or combination across plus and minus signs.
A dependable method
- Look only at addition and subtraction signs not enclosed in grouping symbols.
- Split the expression at those top-level signs.
- Attach each subtraction sign as a negative sign on the following term.
- For each term, identify coefficient and variable part.
- Group like terms only when their complete variable parts match.
Worked example
Representations and interpretation
Algebra tiles or labelled cards can place one term on each card. Factors live within a card; addition or subtraction separates cards. This visual boundary prevents multiplication signs inside a term from being confused with term separators.
Reasoning about variations
The expression $x/3$ is one term because it equals $(1/3)x$. The fraction $(x+1)/3$ contains a grouped numerator and can be rewritten as two terms, $x/3+1/3$, by distribution.
Common mistakes
How to check your work
- Rejoin the identified terms with addition and recover the original expression.
- Substitute simple values before and after combining alleged like terms.
- Compare every variable and exponent in the variable parts.
Practice
- How many terms are in $3x^2-4x+9$?
- Are $5a^2b$ and $-2ab^2$ like terms?
- What is the coefficient of the term $-y$?
Answers and brief solutions
Show answers
- $3$ The top-level terms are $3x^2$, $-4x$, and $9$.
- No Their exponent patterns differ.
- $-1$ $-y=(-1)y$.
Synthesis and transfer
Colour-coding the terms in a polynomial distinguishes coefficients and exponents; only terms with identical variable parts may be collected without changing the expression.
In $4x^2-3x+7$, the signed terms are $4x^2$, $-3x$, and $7$. Treating the minus sign as belonging to the second term makes later collection and substitution unambiguous. Multiplicative pieces inside a term—coefficient, variables, and powers—are factors, not additional terms. Thus $4x^2$ remains one term despite containing three visible symbols. When parentheses are expanded, new additive boundaries may appear; when like terms combine, boundaries may disappear. This grammar explains why cancellation works across factors in a quotient but not across terms joined by addition. Reading expression structure accurately is a prerequisite for nearly every algebraic operation that follows.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
How many terms are in $3x^2-4x+9$?
- The top-level terms are $3x^2$, $-4x$, and $9$.
End of lesson
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