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Marden's theorem

Marden's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Marden's theorem rather than just read about it. In short: In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative. See also geometrical properties of polynomial roots.

Marden's theorem — main illustration
Marden's theorem — illustration

Key takeaways

  • Marden's theorem belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Marden's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Marden's theorem from memory before moving on to harder problems.

Reference excerpt

In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative. See also geometrical properties of polynomial roots.

Statement A cubic polynomial has three zeroes in the complex number plane, which in general form a triangle, and the Gauss–Lucas theorem states that the roots of its derivative lie within this triangle. Marden's theorem states their location within this triangle more precisely:

Suppose the zeroes z1, z2, and z3 of a third-degree polynomial p(z) are non-collinear. There is a unique ellipse inscribed in the triangle with vertices z1, z2, z3 and tangent to the sides at their midpoints: the Steiner inellipse. The foci of that ellipse are the zeroes of the derivative p'(z).

Proof This proof comes from an exercise in Fritz Carlson's book “Geometri” (in Swedish, 1943).

Additional relations between root locations and the Steiner inellipse By the Gauss–Lucas theorem, the root of the double derivative p"(z) must be the average of the two foci, which is the center point of the ellipse and the centroid of the triangle. In the special case that the triangle is equilateral (as happens, for instance, for the polynomial p(z) = z3 − 1) the inscribed ellipse becomes a circle, and the derivative of p has a double root at the center of the circle. Conversely, if the derivative has a double root, then the triangle must be equilateral (Kalman 2008a).

Generalizations A more general version of the theorem, due to Linfield (1920), applies to polynomials p(z) = (z − a)i (z − b)j (z − c)k whose degree i + j + k may be higher than three, but that have only three roots a, b, and c. For such polynomials, the roots of the derivative may be found at the multiple roots of the given polynomial (the roots whose exponent is greater than one) and at the foci of an ellipse whose points of tangency to the triangle divide its sides in the ratios i : j, j : k, and k : i. Another generalization (Parish (2006)) is to n-gons: some n-gons have an interior ellipse that is tangent to each side at the side's midpoint. Marden's theorem still applies: the foci of this midpoint-tangent inellipse are zeroes of the derivative of the polynomial whose zeroes are the vertices of the n-gon.

History Jörg Siebeck discovered this theorem 81 years before Marden wrote about it. However, Dan Kalman titled his American Mathematical Monthly paper "Marden's theorem" because, as he writes, "I call this Marden’s Theorem because I first read it in M. Marden’s wonderful book". Marden (1945, 1966) attributes what is now known as Marden's theorem to Siebeck (1864) and cites nine papers that included a version of the theorem. Dan Kalman won the 2009 Lester R. Ford Award of the Mathematical Association of America for his 2008 paper in the American Mathematical Monthly describing the theorem.

See also Bôcher's theorem for rational functions

References

Kalman, Dan (2008a), "An Elementary Proof of Marden's Theorem", The American Mathematical Monthly, 115 (4): 330–338, doi:10.1080/00029890.2008.11920532, ISSN 0002-9890, S2CID 13222698 Kalman, Dan (2008b), "The Most Marvelous Theorem in Mathematics", Journal of Online Mathematics and Its Applications Linfield, B. Z. (1920), "On the relation of the roots and poles of a rational function to the roots of its derivative", Bulletin of the American Mathematical Society, 27: 17–21, doi:10.1090/S0002-9904-1920-03350-1. Marden, Morris (1945), "A note on the zeroes of the sections of a partial fraction", Bulletin of the American Mathematical Society, 51 (12): 935–940, doi:10.1090/S0002-9904-1945-08470-5 Marden, Morris (1966), Geometry of Polynomials, Mathematical Surveys, vol. 3, Providence, R.I.: American Mathematical Society; reprint of 1949 original publication{{citation}}: CS1 maint: postscript (link); 2005 pbk reprint with corrections Parish, James L. (2006), "On the derivative of a vertex polynomial" (PDF), Forum Geometricorum, 6: 285–288: Proposition 5 Siebeck, Jörg (1864), "Über eine neue analytische Behandlungweise der Brennpunkte", Journal für die reine und angewandte Mathematik, 64: 175–182, ISSN 0075-4102 hathitrust link

Illustrations

Marden's theorem: A triangle and its Steiner inellipse. The zeroes of p(z) are the black dots, and the zeroes of p'(z) are the red dots). The center green dot is the zero of p"(z). Marden's theorem states that the red dots are the foci of the ellipse.
A triangle and its Steiner inellipse. The zeroes of p(z) are the black dots, and the zeroes of p'(z) are the red dots). The center green dot is the zero of p"(z). Marden's theorem states that the red dots are the foci of the ellipse.

Worked examples

Example 1 — a first encounter with Marden's theorem

Start with the simplest possible case. Write down what Marden's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Marden's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Marden's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Marden's theorem

In research
Marden's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Marden's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Marden's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conic sections, Theorems about polynomials, Theorems about triangles, so understanding it makes those chapters shorter.
In everyday life
Look for Marden's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Marden's theorem in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Marden's theorem means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Marden's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Marden's theorem in simple terms?

In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative. See also geometrical properties of polynomial ro…

Why does Marden's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Marden's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Marden's theorem.

Tags

  • Conic sections
  • Theorems about polynomials
  • Theorems about triangles
  • Theorems in complex geometry

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