Term (Algebra)
A term in algebra is a single number, variable, or product of numbers and variables separated from other terms by addition or subtraction.
Definition
A term is one piece of an algebraic expression: a product of constants and variables, separated from other terms by addition or subtraction. A single number (a constant term) or a single variable each count as one term on their own. In a polynomial ring $R[x_1, \ldots, x_n]$, a term is formally a product of a coefficient from ring $R$ and a monomial $x_1^{a_1} \cdots x_n^{a_n}$, and the total degree of a term is the sum of all its exponents; terms are the atomic summands of a polynomial, and the way they are ordered underlies Grobner basis computation and term-order definitions.
Example
In $4x + 9 - 2y$, there are three terms: $4x$, $9$, and $-2y$. In $5x^2 - 3xy + 7$, the term $5x^2$ has coefficient $5$ and variable part $x^2$, while $-3xy$ has coefficient $-3$ and variable part $xy$. In $3x^2y - xy^3 + 2$, the three terms are $3x^2y$ (degree $3$), $-xy^3$ (degree $4$), and $2$ (degree $0$); under lexicographic order with $x > y$, the leading term is $3x^2y$.
Key Insight
Counting the terms in an expression tells you how complex it is: one term is a monomial, two is a binomial, three is a trinomial. The sign in front of a term belongs to that term, so $-3xy$ is a term with a negative coefficient, not a subtraction applied to $3xy$. Choosing a term order (lex, graded lex, graded reverse lex) determines the leading term of each polynomial, controlling the behavior of algorithms like polynomial division and Buchberger's algorithm.