In mathematics a monomial is an algebraic expression made of a single term. Informally, it is a product of a constant factor and one or more variables raised to powers. A monomial contrasts with a general polynomial, which may be a sum of several such terms.
Definition and components
A typical monomial has the form c x_1^{a1} x_2^{a2} ... x_n^{an}, where c is the coefficient (a number) and each exponent ai is a nonnegative integer. The symbols x_i represent variables; a monomial does not allow a variable to appear in a denominator or under a root if that would produce fractional or negative exponents. In particular, a monomial cannot contain a denominator that depends on variables.
Key properties
- The degree of a monomial is the sum of its variable exponents (total degree).
- Constant nonzero numbers are monomials of degree zero; the number zero is sometimes treated separately.
- Two monomials are called like terms if they have the same variable part (same exponents).
Operations and examples
Monomials multiply by multiplying coefficients and adding exponents for each variable: (2x^2)(3x^3)=6x^5. Division of monomials is defined when exponents remain nonnegative: x^5/x^2=x^3. Addition and subtraction combine only like terms; otherwise the result is a polynomial with multiple terms.
Monomials appear throughout algebra and applied mathematics: as building blocks for polynomials, in polynomial factorization, and in modeling power-law relationships. The term itself comes from the prefix mono- meaning "one" combined with a suffix indicating a single term, emphasizing that a monomial contains exactly one algebraic term.