# What is the definition of coefficient?

How to define coefficient? And what does it mean when someone says: put the $i$th term to be the coefficient of $x^i$

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It means the constant multiplying the term $x^i$.
E.g., If f(x) = $x^3 + 2x^2 + x + 9$, then the coefficient of $x^3 = 1\cdot x^3$ is $1$, the coefficient of the term $x^2$ is $2$, the coefficient of the term $x = x^1$ is $1$, and the coefficient of the term $x^0 = 1$ is $9$.
So "put the $i$th term to be the coefficient of $x^i$" would require knowledge of the domain of $i$. Suppose the $4$th term is $3$. Then we would multiply $x^4$ by $3$, to get $3x^4$.
Note: In a polynomial such as $4x^2 - 1$, the coefficient of the "absent" term "$x$" can be said to be $0$.
If the 8th term is $8$, then you'd have $8x^8$. But if $i = 8$, then you'd need to look at the $8$th term. If the 8th term is, say $7$, then you'd multiply $7\cdot x^8$. – amWhy Mar 9 '13 at 19:44