Polynomial function
High school
A polynomial function is a function whose output is given by a polynomial in its input.
Undergraduate
Consider a field $F$ and a function $f : F \to F$. The function $f$ is a polynomial function if, for some $a_0, \dots, a_n \in F$,
\[f(x) = a_0 + a_1 x + \dots + a_n x^n \quad \text{for all } x \in F.\]Two polynomial functions are equal when they give the same output for every input.
Examples
Examples
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$f(x) = x^2 - 1$, for $x \in \reals$
High school
$f(x) = x^2 - 1$ for $x \in \reals$ is a polynomial function because it evaluates a polynomial at each real number.
Non-examples
Things that may be mistaken for examples.
-
$x^2 - 1$
High school
$x^2 - 1$ is not a polynomial function, because it is an expression. It defines a function only once we say what $x$ ranges over.
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$x \mapsto \sin x$
High school
$\sin x$ is not a polynomial function. It has infinitely many roots, but a nonzero polynomial has only finitely many.
Properties
- Over an infinite field, such as $\reals$ or $\mathbb{C}$, different polynomials give different functions, so polynomials and polynomial functions are usually identified.
- Over a finite field, different polynomials can give the same function. Over $\setdef{0, 1}$ with arithmetic modulo $2$, the polynomial $x^2 + x$ gives the zero function.