Polynomial Function

analysis

Definition

Polynomial Function

A function $f:\mathbb{R}\to \mathbb{R}$ of form:

$$f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$$

is called polynomial function.

Properties

Bijectivity

Bijectivity

A polynomial function $f_n:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $f_n(x)=x^n$ and exponent $n\in \mathbb{N}\setminus \{0\}$ is bijective.

Continuity

Continuity of Polynomial Functions

Polynomial functions are always continuous.

Degree

Degree

The degree of a polynomial function $f(x)$ is the highest exponent of the variable $x$.

For example, the degree of the following function is $4$ since the highest exponent of $x$ is 4:

$$f(x)=4x^4+3x^3+12$$

The degree of a polynomial function states the following properties about the function itself:

• Maximum number of roots = degree
• Maximum number of extrema = degree - 1
• Maximum number of turning points = degree - 2