Polynomial function

Remarks

The coefficients are unique: if two such formulas give the same function, their difference is a polynomial with infinitely many roots, so all its coefficients are $0$. So each polynomial function comes from exactly one element of the polynomial ring $\R[x]$, and its degree is well defined. The same definition works on an interval of $\R$, or with $\R$ replaced by $\C$. Every polynomial function is continuous and differentiable, and its derivative is the polynomial function $a_1 + 2 a_2 x + \dots + n a_n x^{n-1}$.

Example: $f(x) = (x - 1)^2$ is a polynomial function, since $(x - 1)^2 = 1 - 2x + x^2$ for all $x$, with $n = 2$, $a_0 = 1$, $a_1 = -2$, $a_2 = 1$.

Non-example: $f(x) = \abs{x}$ is not a polynomial function. If it were equal to a polynomial $p$, then $p(x) - x$ would vanish for all $x > 0$, so $p(x) = x$; but then $p(-1) = -1 \ne \abs{-1}$.