Lill's Construction for a Depressed Cubic Polynomial

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

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Consider the trigonometric identity for in the form

.

Substitute and to get the polynomial.

The equation

has the solutions

,

,

.

By Lill's construction,

,

so given , we must find such that (the length of the thick red segment is 0).

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Contributed by: Izidor Hafner (September 2017)
Open content licensed under CC BY-NC-SA


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Details

Since we are mainly interested in zeros of the polynomial , it looks like this Demonstration is less general than the so-called depressed equation . But Vieta showed that in the irreducible case, using the substitution gives the equation [2, p. 133].

References

[1] D. Kurepa, Higher Algebra, Book 2 (in Croatian), Zagreb: Skolska knjiga, 1965 pp. 1072–1074.

[2] G. E. Martin, Geometric Constructions, New York: Springer, 1998.



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