monomials

A monomial is a single term in algebra that consists of a coefficient multiplied by nonnegative integer powers of variables. In a multivariable context, a monomial has the form a x1^a1 x2^a2 ... xn^an, where each exponent ai is a nonnegative integer and a is a coefficient. In common practice, the coefficient is typically nonzero; the zero term is not considered a monomial, and the zero polynomial is treated separately.

Examples include 7, -3x^2, 4xy^3, and 5x^4y^2z. In one variable, monomials are expressions like 6x^3 or -2x. In several variables, exponents refer to the corresponding variable, so x^2y^4z^1 is a monomial.

Degree and structure: The total degree of a monomial is the sum of its exponents. For example, the degree of 6x^3y^2 is 5. The concept can be extended to polynomials by combining many monomials, as polynomials are sums of monomials.

Operations: Monomials obey basic exponent rules. Multiplication combines coefficients and adds exponents (e.g., (2x^2y)(3xy^3) = 6x^3y^4). Division subtracts exponents, and raising a monomial to a power multiplies the exponents (e.g., (ax^i)^n = a^n x^{in}). Factoring a polynomial involves expressing it as a sum of monomials or as a product of monomials and other factors.

Monomials form the building blocks of polynomials and are used in various areas of algebra, including polynomial long division, factoring, and algorithmic computation. They are also ordered and graded in many algebraic structures, according to degree or other criteria.