Coefficient

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In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor.^[1] It may also be a constant with units of measurement, in which it is known as a constant multiplier.^[1] In general, coefficients may be any expression (including variables such as a, b and c).^[2][1] When the combination of variables and constants is not necessarily involved in a product, it may be called a parameter.^[1] For example, the polynomial ${\displaystyle 2x^{2}-x+3}$ has coefficients 2, −1, and 3, and the powers of the variable ${\displaystyle x}$ in the polynomial ${\displaystyle ax^{2}+bx+c}$ have coefficient parameters ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$.

A 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 ⋅ x⁰. 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 equations, 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 with constant coefficient, the constant coefficient term is generally not assumed to be a constant function.