Polynomial

A finite linear combination of powers of a variable with real coefficients.

Polynomial

A polynomial (in one real variable) is a function $p:\mathbb{R}\to\mathbb{R}$ of the form

p(x)=a_0+a_1x+a_2x^2+\cdots+a_n x^n

for some integer $n\ge 0$ and coefficients $a_0,\dots,a_n\in\mathbb{R}$ . If $a_n\ne 0$ , the integer $n$ is the degree of $p$ .

Polynomials are basic examples of smooth functions (they have derivatives of all orders ) and they define continuous functions on any interval, so they sit inside the space of continuous functions on that interval. Collections of polynomials often form a subalgebra of continuous functions .

Examples:

$p(x)=x^3-2x+5$ is a polynomial of degree $3$ .
$p(x)=7$ is a constant polynomial (degree $0$ ), and $p(x)=0$ is the zero polynomial.