WOLFRAM|DEMONSTRATIONS PROJECT

Lill's Construction for a Depressed Cubic Polynomial

set polynomial

b

-2

move

x

1.5

show labels

show axes

show grid lines

P(x) =

3

x

-3x+2

This Demonstration shows a graphic check of Lill's construction of a polynomial of the form

3

x

-3x-b

where

-2≤b≤2

.

Consider the trigonometric identity for

cos(3α)

in the form

3

(2cosα)

-3(2cosα)-2cos(3α)=0

.

Substitute

x=2cosα

and

b=2cos(3α)

to get the polynomial.

The equation

P(x)=

3

x

-3x-2cos(3α)=0

has the solutions

2cos(α)

,

2cosα+

2π

3

,

2cosα+

4π

3

.

By Lill's construction,

P(x)=

L

4

L'

3



,

so given

b

, we must find

x

such that



L

4

L'

3

=0

(the length of the thick red segment is 0).