# 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