WOLFRAM|DEMONSTRATIONS PROJECT

Geometry of Quartic Polynomials

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Tetrahedron Rotation
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Display Elements
lines from projected vertices
first derivative triangle
conjugatefirst derivative triangle
second derivative roots
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
Given a quartic
p(x)
with four real roots (at least two distinct), those roots are the first coordinate projections of a regular tetrahedron in
3

. That tetrahedron has a unique inscribed sphere, which projects onto an interval whose endpoints are the two roots of
p''(x)
. Let
r
,
s
,
t
be the roots of
p'(x)
; then the points
r,
s-t
3
,
s,
t-r
3
,
t,
r-s
3
form an equilateral triangle whose vertices project on the critical points of the quartic
p(x)
(this is the "first derivative triangle"). A similar triangle is formed by negating the
y
coordinates of these points ("conjugate first derivative triangle"). This application is relevant to the following so far open conjecture: there does not exist a quartic polynomial
p
with four distinct rational roots such that
p'
,
p''
, and
p'''
all have rational roots.