Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression. In the latter case, the variables appearing in the coefficients are often called parameters, and must be clearly distinguished from the other variables.

For example, in

${\displaystyle 7x^{2}-3xy+1.5+y,}$

the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant coefficient. The final term does not have any explicitly written coefficient factor that would not change the term; the coefficient is taken to be 1.

Often coefficients are numbers as in this example, although they could be parameters of the problem or any expression in these parameters. In such a case one must clearly distinguish between symbols representing variables and symbols representing parameters. Following René Descartes, the variables are often denoted by x, y, ..., and the parameters by a, b, c, ..., but it is not always the case. For example, if y is considered as a parameter in the above expression, the coefficient of x is −3y, and the constant coefficient is 1.5 + y.

When one writes

${\displaystyle ax^{2}+bx+c,}$

it is generally supposed that x is the only variable and that a, b and c are parameters; thus the constant coefficient is c in this case.

Similarly, any polynomial in one variable x can be written as

${\displaystyle a_{k}x^{k}+\dotsb +a_{1}x^{1}+a_{0}}$

for some positive integer ${\displaystyle k}$, where ${\displaystyle a_{k},\dotsc ,a_{1},a_{0}}$ are coefficients; to allow this kind of expression in all cases one must allow introducing terms with 0 as coefficient. For the largest ${\displaystyle i}$ with ${\displaystyle a_{i}\neq 0}$ (if any), ${\displaystyle a_{i}}$ is called the leading coefficient of the polynomial. So for example the leading coefficient of the polynomial

${\displaystyle \,4x^{5}+x^{3}+2x^{2}}$

is 4.

Some specific coefficients that occur frequently in mathematics have received a name. For example, the binomial coefficients occur in the expanded form of ${\displaystyle (x+y)^{n}}$, and are tabulated in Pascal's triangle.