Coefficient

For other uses, see Coefficient (disambiguation).

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 in any case does not involve any variable of the expression. For instance 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. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term. 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 clarify which symbols represent variables and which ones represent parameters. Following 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 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.

Specific coefficients arise in mathematical identities, such as the binomial theorem which involves binomial coefficients; these particular coefficients are tabulated in Pascal's triangle.

Linear algebra

In linear algebra, the leading coefficient (also leading entry) of a row in a matrix is the first nonzero entry in that row. So, for example, given

${\displaystyle M={\begin{pmatrix}1&2&0&6\\0&2&9&4\\0&0&0&4\\0&0&0&0\end{pmatrix}}}$.

The leading coefficient of the first row is 1; 2 is the leading coefficient of the second row; 4 is the leading coefficient of the third row, and the last row does not have a leading coefficient.

Though coefficients are frequently viewed as constants in elementary algebra, they can be variables more generally. For example, the coordinates ${\displaystyle (x_{1},x_{2},\dotsc ,x_{n})}$ of a vector ${\displaystyle v}$ in a vector space with basis ${\displaystyle \lbrace e_{1},e_{2},\dotsc ,e_{n}\rbrace }$, are the coefficients of the basis vectors in the expression

${\displaystyle v=x_{1}e_{1}+x_{2}e_{2}+\dotsb +x_{n}e_{n}.}$