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. 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.