# Coefficient

In mathematics, a coefficient is a multiplicative factor in some term o an expression (or o a series); it is uisually a nummer, but in ony case daes nae involve ony variables o the expression. For instance in

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

the first twa terms respectively hae the coefficients 7 an −3. The third term 1.5 is a constant. The final term daes nae hae ony explicitly written coefficient, but is considered tae hae coefficient 1, syne multiplyin bi that factor would nae chynge the term. Eften coefficients are nummers as in this example, altho thay could be parameters o the problem, as a, b, an c in

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

when it is unnerstuid that these are nae considered as variables. Thus a polynomial in ane variable x can be written as

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

for some integer ${\displaystyle k}$, whaur ${\displaystyle a_{k},\dotsc ,a_{1},a_{0}}$ are coefficients; tae allae this kind o expression in aw cases ane must allae introducing terms wi 0 as coefficient. For the lairgest ${\displaystyle i}$ with ${\displaystyle a_{i}\neq 0}$ (if any), ${\displaystyle a_{i}}$ is cried the leadin coefficient o the polynomial. So for example the leadin coefficient o 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 pairticular coefficients are tabulatit in Pascal's triangle.