Monimial

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

1. A monomial, also called power product, is a product of powers of variables (a) with nonnegative integer (a) exponents, or, in other words, a product of variables, possibly with repetitions. For example, \(x^{2}yz^{3} = xxyzzz\)https://wikimedia.org/api/rest_v1/media/math/render/svg/020356e343338ea14d3c64e0dc1f049e6cebb6a6 (a) is a monomial. The constant \(1\)https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf (a) is a monomial, being equal to the empty product (a) and to \(x^{0}\)https://wikimedia.org/api/rest_v1/media/math/render/svg/1871ffeb57c11624b375dbb7157d5887c706eb87 (a) for any variable \(x\)https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4 (a). If only a single variable \(x\)https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4 (a) is considered, this means that a monomial is either \(1\)https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf (a) or a power \(x^{n}\)https://wikimedia.org/api/rest_v1/media/math/render/svg/150d38e238991bc4d0689ffc9d2a852547d2658d (a) of \(x\)https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4 (a), with \(n\)https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b (a) a positive integer. If several variables are considered, say, \(x,y,z,\)https://wikimedia.org/api/rest_v1/media/math/render/svg/d08d690d7e19ea7aee8574fc6abd6a15d97fa026 (a) then each can be given an exponent, so that any monomial is of the form \(x^{a}y^{b}z^{c}\)https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc6a3a6ad33395f3118d5cfbea58c9ac722309b (a) with \(a,b,c\)https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d (a) non-negative integers (taking note that any exponent \(0\)https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950 (a) makes the corresponding factor equal to \(1\)https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf (a)).
2. A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient (a) of the monomial. A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is \(1\)https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf (a). For example, in this interpretation \(- 7x^{5}\)https://wikimedia.org/api/rest_v1/media/math/render/svg/c89cb86c7f43c2860265e1ff3ba8fb505cc9076a (a) and \((3 - 4i)x^{4}yz^{13}\)https://wikimedia.org/api/rest_v1/media/math/render/svg/53e747f495086586d7eba3e54897e5432208d9aa (a) are monomials (in the second example, the variables are \(x,y,z,\)https://wikimedia.org/api/rest_v1/media/math/render/svg/d08d690d7e19ea7aee8574fc6abd6a15d97fa026 (a) and the coefficient is a complex number (a)).

(“Monomial” 2022)

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