Editing Coefficient (section)

{{short description|Multiplicative factor in a mathematical expression}}
{{Other uses}}
{{refimprove|date=May 2017}}

In [[mathematics]], a '''coefficient''' is a [[Factor (arithmetic)|multiplicative factor]] involved in some [[Summand|term]] of a [[polynomial]], a [[series (mathematics)|series]], or any other type of [[expression (mathematics)|expression]]. It may be a [[Dimensionless quantity|number without units]], in which case it is known as a '''numerical factor'''.<ref name=80000-1:2009>{{cite web|title=ISO 80000-1:2009|url=https://www.iso.org/standard/30669.html|publisher=[[International Organization for Standardization]]|access-date=2019-09-15}}</ref> It may also be a [[constant (mathematics)|constant]] with [[units of measurement]], in which it is known as a '''constant multiplier'''.<ref name=80000-1:2009/> In general, coefficients may be any [[mathematical expression|expression]] (including [[Variable (mathematics)|variables]] such as {{mvar|a}}, {{mvar|b}} and {{mvar|c}}).<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Coefficient |url=https://mathworld.wolfram.com/Coefficient.html |access-date=2020-08-15 |website=mathworld.wolfram.com |language=en}}</ref><ref name=80000-1:2009/> When the combination of variables and constants is not necessarily involved in a [[product (mathematics)|product]], it may be called a ''[[parameter]]''.<ref name=80000-1:2009/>  
For example, the polynomial <math>2x^2-x+3</math> has coefficients 2, −1, and 3, and the powers of the variable <math>x</math> in the polynomial <math>ax^2+bx+c</math> have coefficient parameters <math>a</math>, <math>b</math>, and <math>c</math>.

A {{vanchor|Constant coefficient|text='''constant coefficient'''}}, also known as '''constant term''' or simply '''constant''', is a quantity either implicitly attached to the [[zeroth power]] of a variable or not attached to other variables in an expression; for example, the constant coefficients of the expressions above are the number 3 and the parameter ''c'', involved in 3{{=}}''c''{{sdot}}''x''<sup>0</sup>. 
The coefficient attached to the highest degree of the variable in a polynomial of one variable is referred to as the '''leading coefficient'''; for example, in the example expressions above, the leading coefficients are 2 and ''a'', respectively.

In the context of [[differential equation]]s, these equations can often be written in terms of polynomials in one or more unknown functions and their derivatives. In such cases, the coefficients of the differential equation are the coefficients of this polynomial, and these may be non-constant functions. A coefficient is a ''constant coefficient'' when it is a [[constant function]]. For avoiding confusion, in this context a coefficient that is not attached to unknown functions or their derivatives is generally called a ''constant term'' rather than a constant coefficient. In particular, in a [[linear differential equation#Non-homogeneous equation with constant coefficients|linear differential equation with constant coefficient]], the constant coefficient term is generally not assumed to be a constant function.