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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).
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