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