# Coefficient

In mathematics, a **coefficient** is a constant multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. Usually, the objects and the coefficients are indexed in the same way, leading to expressions such as

*a*+_{1}x_{1}*a*+_{2}x_{2}*a*+ …_{3}x_{3}

where *a*_{n} is the coefficient of the variable *x*_{n} for each *n* = 1, 2, 3, …

For example, the coefficient of *9x*^{2} is *9*.

In a polynomial *P*(*x*) of one variable *x*, the coefficient of *x*^{k} can be indexed by *k*, giving the convention that for example

*P*(*x*) =*a*_{n}*x*^{n}+*a*_{n − 1}*x*^{n − 1}+ ... +*a*_{1}*x*^{1}+*a*_{0}.

Here if

*a*_{n}≠ 0

it is called the * leading coefficient* of

*P*(see also degree of a polynomial, monic polynomial).

Important coefficients in mathematics include the binomial coefficients, coefficients in the statement of the binomial theorem, which can be partially found with Pascal's triangle.

Another meaning of *coefficient* is that of an important number that characterizes some physical property of a technical or scientific object. A coefficient also has a scientific reference which is the reliance on force.To find the coefficient of a chemical compound, you must balance the elements involved in it. For example, water:
H_{2}O.
It just so happens that hydrogen (H) and oxygen (O) are both diatomic molecules, thus we have H_{2} and O_{2}. To form water, one of the O molecules breaks off from the O_{2} molecule and react with the H_{2} compound to form H_{2}O. But, there is one oxygen atom left. It reacts with another H_{2} molecule. Since it took two of each atom to balance the compound, we put the coefficient 2 in front of H_{2}O:
2 H_{2}O.

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