Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:

1. A monomial, also called a power product or primitive monomial,^[1] is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions.^[2] For example, ${\displaystyle x^{2}yz^{3}=xxyzzz}$ is a monomial. The constant ${\displaystyle 1}$ is a primitive monomial, being equal to the empty product and to ${\displaystyle x^{0}}$ for any variable ${\displaystyle x}$. If only a single variable ${\displaystyle x}$ is considered, this means that a monomial is either ${\displaystyle 1}$ or a power ${\displaystyle x^{n}}$ of ${\displaystyle x}$, with ${\displaystyle n}$ a positive integer. If several variables are considered, say, ${\displaystyle x,y,z,}$ then each can be given an exponent, so that any monomial is of the form ${\displaystyle x^{a}y^{b}z^{c}}$ with ${\displaystyle a,b,c}$ non-negative integers (taking note that any exponent ${\displaystyle 0}$ makes the corresponding factor equal to ${\displaystyle 1}$).
2. A monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial.^[1] A primitive monomial is a special case of a monomial in this second sense, where the coefficient is ${\displaystyle 1}$. For example, in this interpretation ${\displaystyle -7x^{5}}$ and ${\displaystyle (3-4i)x^{4}yz^{13}}$ are monomials (in the second example, the variables are ${\displaystyle x,y,z,}$ and the coefficient is a complex number).

In the context of Laurent polynomials and Laurent series, the exponents of a monomial may be negative, and in the context of Puiseux series, the exponents may be rational numbers.

In mathematical analysis, it is common to consider polynomials written in terms of a shifted variable ${\displaystyle {\bar {x}}=x-c}$ for some constant ${\displaystyle c}$ rather than a variable ${\displaystyle x}$ alone, as in the study of Taylor series.^[3][4] By a slight abuse of notation, monomials of shifted variables, for instance ${\displaystyle 2{\bar {x}}^{3}=2(x-c)^{3},}$ may be called monomials in the sense of shifted monomials or centered monomials, where ${\displaystyle c}$ is the center or ${\displaystyle -c}$ is the shift.

Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the prefix "bi-" (two in Latin), a monomial should theoretically be called a "mononomial". "Monomial" is a syncope by haplology of "mononomial".^[5]