Lill's Construction for a Depressed Cubic Polynomial
Lill's Construction for a Depressed Cubic Polynomial
This Demonstration shows a graphic check of Lill's construction of a polynomial of the form -3x-b where .
3
x
-2≤b≤2
Consider the trigonometric identity for in the form
cos(3α)
3
(2cosα)
Substitute and to get the polynomial.
x=2cosα
b=2cos(3α)
The equation
P(x)=-3x-2cos(3α)=0
3
x
has the solutions
2cos(α)
2cosα+
2π
3
2cosα+
4π
3
By Lill's construction,
P(x)=
L
4
L'
3
so given , we must find such that (the length of the thick red segment is 0).
b
x
=0
L
4
L'
3