Polynomial

A Polynomial is an expression that:

Is made up of 2 or more algebraic terms
Especially the sum of several terms that contain different powers of the same variable(s)

Examples of Polynomials:

Powers of the variable x multiplied by real numbers are called monomials (circled in darker green in the graph). These are added together as a sum to form the polynomial (in the square green box).

In a polynomial the multiples of several power functions are added, whose exponents come from the set {N} (natural numbers).

Examples for Polynomials:

$x^{3} - 2 x^{2} + x - 4$
$8 x^{5} + 3 x^{3} - x^{2} + 12$
$4 x^{7} + x^{5} - 2 x^{4} + x - 6$

Mathematical Definition

Mathematically, a polynomial is a term that can be written in the following form:

$a_{n} \cdot x^{n} + a_{n - 1} \cdot x^{n - 1} + \dots + a_{1} \cdot x + a_{0}$

Where x is the variable, $a_{n}, a_{n - 1}, \dots a_{1}, a_{0}$ are real numbers( $a_{n} \neq 0$ ), and $n$ is a natual number.

Example

$4 x^{7} + x^{5} - 2 x^{4} + x - 6$

$\Rightarrow$ here is $n = 7$ , $a_{7} = 4$ , $a_{5} = 1$ , $a_{4} = - 2$ , $a_{1} = 1$ , $a_{0} = - 6$

Ordered Polynomials

Usually a polynomial is written down in an ordered way. A polynomial is called "ordered" if the polynomial is summarized and sorted by falling exponents.

So not in the form of $2 x^{2} + 1 - x^{7}$ ,but like this $- x^{7} + 2 x^{2} + 1$ .

Degree of the polynomial

The degree of the polynomial is the highest occurring power.

So we here we have a few examples…

$x^{3} - 2 x^{2} + x - 4$ den Grad $3$ ,
$8 x^{5} + 3 x^{3} - x^{2} + 12$ den Grad $5$ und
$4 x^{7} + x^{5} - 2 x^{4} + x - 6$ den Grad $7$ .

The function of the Polynomial:

A function $f$ : $x \mapsto f (x)$ , written as $f (x)$ is a polynomial, is called an integer function or polynomial function.

You can find more information about the features and properties of such a function in the article : Integer functions (polynomial functions).

This content is licensed under
CC BY-SA 4.0 → What does that mean? serlo.org