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Title: Marden's Theorem
Book
Description: Marden's
theorem concerns polynomials over the complex numbers.
As a specific case, consider the polynomial
p(z) = (z-a)(z-b)(z-c),
where a, b, and c are fixed complex numbers, and z is a complex variable.
We know that p has roots at a, b, and c, which we visualize as three points
in the complex plane. If those points are not collinear, they define a
triangle within which it is possible to inscribe an ellipse, tangent to
the sides at their midpoints.
Marden's theorem says that the foci of the ellipse are precisely the roots
of the derivative, p'(z). In this WorkBook, you may experiment with this,
and a more general version of Marden's Theorem. And you may review properties
of complex numbers and the complex plane, both visually and algebraically.
This WorkBook will also give you the opportunity to explore the geometry
of ellipses inscribed in triangles.
This is a "hands on" opportunity to investigate some beautiful correlations
between algebra and geometry. At every step, the book invites you to ask
questions, and to see for yourself what the answers to those questions
are. You will enjoy this one!
Author: Dan Kalman
Suggested Use: Experiment with the geometry of cubic polynomial equations.
Topics: complex numbers, complex arithmetic, cubic polynomials, geometry, Marden's theorem
Number of Pages: 9
Animation: Yes
Grade
Level: 