Coefficient

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In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or an expression. It may be a number (dimensionless), 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 ${\displaystyle c}$.

The constant coefficient, also known as constant term or simply constant, is the quantity not attached to variables in an expression or implicitely attached to the zeroth power of a variable; 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 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.