Monomial
AlgebraA 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 just one term. It can be a number, a variable, or a number multiplied by one or more variables.
Example
$5$, $3x$, $-7x^2$, $4xy$, and $2x^3y^2$ are all monomials. Each has exactly one term with no addition or subtraction.
Key Insight
"Mono" means one. A monomial has one term, a binomial has two, and a trinomial has three.
Definition
A monomial is a product of a constant coefficient and one or more variables each raised to a non-negative integer power. The degree of a monomial is the sum of all variable exponents. Monomials are the individual terms that make up polynomials.
Example
$6x^2y^3$ is a monomial with coefficient $6$, and degree $2 + 3 = 5$. Multiplying monomials: $(3x^2)(4x^3) = 12x^5$ (multiply coefficients, add exponents).
Key Insight
When multiplying monomials, coefficients multiply and exponents add. This is the product rule for exponents applied directly.
Definition
A monomial in $n$ variables $x_1,\ldots,x_n$ is a product $x_1^{a_1} \cdot x_2^{a_2} \cdot \ldots \cdot x_n^{a_n}$ (times a constant) where each $a_i$ is a non-negative integer. Monomials form a basis for the polynomial ring $F[x_1,\ldots,x_n]$ as a vector space over $F$. A monomial order (e.g., lex, grlex, grevlex) is needed to define leading terms and Grobner bases for multivariable polynomials.
Example
In $F[x,y,z]$, the monomial $x^2yz^3$ has degree $6$. Under lex order with $x > y > z$, $x^2yz^3 > xy^4z^2$ because the x-exponent ($2 > 1$) is compared first.
Key Insight
Monomial orders are necessary to define division algorithms in polynomial rings over multiple variables, enabling the computation of Grobner bases used in solving systems of polynomial equations.