# Definition: Complex Polynomials

Let $$a_0,a_1,\ldots,a_n$$ be complex numbers with $$a_n\neq 0$$. A complex polynomial (or just a polynomial) is a function. $p:=\cases{ \mathbb C\to\mathbb C\\ z\to p(z):=a_nz^n + \ldots + a_1z + a_0\\ }$

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 (complex) coefficients $$a_j=x_j+iy_j$$, $$j=0,1,2,3$$, by manipulating their real (respectively imaginary) parts $$x_j$$ (respectively $$y_j$$). You can then study the behavior of the resulting polynomials of a degree up to $$3$$. The images of a circle and a segment in the complex plane are shown. The initial polynomial (when the Reset button is pressed) is of degree $$0$$ ("constant polynomial").

$w:=a_3z^3+a_2z^2+a_1z+a_0$

$z$-plane

$w$-plane

Definitions: 1
Proofs: 2 3
Propositions: 4

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