Term (Algebra)
Pre-AlgebraA 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. Terms are separated by plus or minus signs. A term can be a number, a variable, or a number multiplied by a variable.
Example
In $4x + 9 - 2y$, there are three terms: $4x$, $9$, and $-2y$. Each piece separated by $+$ or $-$ is its own term.
Key Insight
Counting the terms in an expression helps you know how complex it is: one term is a monomial, two is a binomial, three is a trinomial.
Definition
A term is a product of constants and variables. Terms are the parts of a polynomial or expression connected by addition or subtraction. A single number (constant term) and a single variable are each one-term expressions.
Example
In $5x^2 - 3xy + 7$, the terms are $5x^2$, $-3xy$, and $7$. The term $5x^2$ has coefficient $5$ and variable part $x^2$. The term $-3xy$ has coefficient $-3$ and variable part $xy$.
Key Insight
The sign in front of a term belongs to that term. $-3xy$ is a term with a negative coefficient, not a subtraction operation applied to $3xy$.
Definition
In a polynomial ring $R[x_1, \ldots, x_n]$, a term is a product of a coefficient from ring $R$ and a monomial $x_1^{a_1} \cdots x_n^{a_n}$. The total degree of a term is the sum of all exponents. Terms are the atomic summands of a polynomial, and their set structure is used in Grobner basis computation and term-order definitions.
Example
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
Choosing a term order (lex, graded lex, graded reverse lex) determines the leading term of each polynomial, which controls the behavior of algorithms like polynomial division and Buchberger's algorithm.