Definition: Polynomials

Let \(a_0,a_1,\ldots,a_n\) be real numbers with \(a_n\neq 0\). A real polynomial (or just a polynomial) is a function. \[p:=\begin{cases} \mathbb R&\to\mathbb R\\ x&\to p(x):=a_nx^n + \ldots + a_1x + a_0\\ \end{cases}\]

The numbers \(a_0,a_1,\ldots,a_n\) are called the coefficients of the polynomial. The highest number \(n\), for which the coefficient \(a_n\neq 0\), is called the degree of the polynomial.

In the following interactive figure, you can drag the sliders to manipulate the values of the coefficients \(a_0,a_1,\ldots,a_5\) and see the behavior of resulting polynomials of the degree up to \(5\). The initial polynomial (when the Reset button is pressed) is of degree \(0\).

  1. Proposition: Limits of Polynomials at Infinity
  2. Proposition: Eveness (Oddness) of Polynomials

Branches: 1
Chapters: 2
Corollaries: 3 4
Definitions: 5 6 7 8
Parts: 9
Problems: 10
Proofs: 11 12 13
Propositions: 14 15 16 17
Sections: 18
Solutions: 19

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