coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as a, b and c). When the coefficients are themselves variables, they may also be called parameters.
For example, the polynomial



2

x

2


−
x
+
3


{\displaystyle 2x^{2}-x+3}
has coefficients 2, −1, and 3, and the powers of the variable



x


{\displaystyle x}
in the polynomial



a

x

2


+
b
x
+
c


{\displaystyle ax^{2}+bx+c}
have coefficient parameters



a


{\displaystyle a}
,



b


{\displaystyle b}
, and



c


{\displaystyle c}
.
The constant coefficient (or constant term) is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the number 3 and the parameter c, respectively.
The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and a, respectively.
In the context of differential equations, an equation can often be written as equating to zero a polynomial in the unknown functions and their derivatives. In this case, the coefficients of the differential equation are the coefficients of this polynomial, and are generally non-constant functions. A coefficient is a constant coefficient when it is a constant function. For avoiding confusion, the coefficient that is not attached to unknown functions and their derivative is generally called the constant term rather the constant coefficient. In particular, in a linear differential equation with constant coefficient, the constant term is generally not supposed to be a constant function.

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