Monomial

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

For the first definition, a monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. The constant 1 is a monomial, being equal to the empty product and

x

⁰ for any variable

x

. If only a single variable

x

is considered, this means that a monomial is either 1 or a power

x n

x

, with

n

a positive integer. If several variables are considered, say,

x, y, z,

then each can be given an exponent, so that any monomial is of the form

x^a y^b z^c

with

a,b,c

non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1).

For the second definition, a monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is also a monomial in the second sense, because the multiplication by 1 is allowed. For example, in this interpretation

-7x^5

and

(3-4i)x^4yz^{13}

are monomials (in the second example, the variables are

x, y, z,

and the coefficient is a complex number).

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