Monomial
A monomial is a polynomial with exactly one term, consisting of a coefficient multiplied by variables raised to non-negative integer powers.
Formula
a \cdot x^n
Definition
A monomial is a polynomial made of exactly one term: a number, a variable, or a number multiplied by one or more variables, each raised to a non-negative integer power. Its degree is the sum of all the variable exponents in that single term, and monomials are the individual building blocks that combine to form polynomials; when multiplying monomials, coefficients multiply and exponents add, the product rule for exponents applied directly. In $n$ variables $x_1,\ldots,x_n$, a monomial is a product $x_1^{a_1} \cdot x_2^{a_2} \cdot \ldots \cdot x_n^{a_n}$ (times a constant), and monomials form a basis for the polynomial ring $F[x_1,\ldots,x_n]$ as a vector space over $F$; a monomial order (such as lex, grlex, or grevlex) is needed to define leading terms and Grobner bases for multivariable polynomials.
Example
$5$, $3x$, $-7x^2$, $4xy$, and $2x^3y^2$ are all monomials, each with exactly one term and no addition or subtraction. $6x^2y^3$ has coefficient $6$ and degree $2 + 3 = 5$; multiplying $(3x^2)(4x^3) = 12x^5$. In $F[x,y,z]$, the monomial $x^2yz^3$ has degree $6$, and under lex order with $x > y > z$, $x^2yz^3 > xy^4z^2$ because the x-exponent ($2 > 1$) is compared first.
Key Insight
"Mono" means one: a monomial has one term, a binomial has two, and a trinomial has three. Monomial orders become necessary once you move to multiple variables, since they define how division algorithms work in polynomial rings, enabling the computation of Grobner bases used to solve systems of polynomial equations.