Elite Math Program

Tag: derivative

  • Marden’s Theorem: How the Derivative of a Cubic Finds an Ellipse

    What can the derivative of a polynomial tell us about geometry? For a cubic polynomial with three complex roots, the answer is surprisingly precise: the derivative locates the two foci of a particular ellipse.

    Let

    p(z) = (z−z1) (z−z2) (z−z3),

    where z1, z2, z3 are three distinct, noncollinear complex numbers.

    Every complex number can be represented as a point in the complex plane. Therefore the three roots of p become three points, and those three points form a triangle.

    Now differentiate. Since p is cubic, p′(z) is quadratic and has two roots, counted with multiplicity. Call them w1 and w2.

    Where are these two critical points located, and what do they have to do with the triangle formed by the original three roots?

    Three complex roots z1, z2, and z3 plotted as points in the complex plane and joined to form a triangle.
    Figure 1. Three noncollinear roots z₁, z₂, and z₃ of a cubic polynomial determine a triangle in the complex plane.

    What does the derivative know about the triangle?

    There is already a classical theorem that gives us some information. The Gauss–Lucas theorem states that the zeros of the derivative of a polynomial lie in the convex hull of the zeros of the polynomial.

    For our cubic, the convex hull of the three roots is simply the triangle with vertices z1, z2, z3. Therefore the two zeros of p′ must lie inside this triangle.

    That is already an interesting connection between differentiation and geometry. But for a cubic, something much stronger is true.

    Triangle formed by three complex roots z1, z2, and z3, with the two derivative zeros w1 and w2 shown inside the triangle.
    Figure 2. By the Gauss–Lucas theorem, the two zeros w₁ and w₂ of the derivative lie inside the triangle determined by the three roots of the cubic.

    Marden’s theorem

    Recall the Steiner inellipse of a triangle: the unique ellipse tangent to the three sides at their midpoints.

    We discussed this ellipse and an area characterization of it in The Steiner Inellipse and a Surprising Area Characterization .

    Marden’s theorem reveals a completely different way in which the same ellipse appears.

    Marden’s Theorem. Let

    p(z) = (z−z1) (z−z2) (z−z3),

    where the three roots are noncollinear. Then the two zeros of p′(z) are exactly the two foci of the Steiner inellipse of the triangle whose vertices are z1, z2, z3.

    This is much stronger than Gauss–Lucas. Gauss–Lucas tells us that the critical points are somewhere inside the triangle. Marden’s theorem identifies their exact geometric meaning.

    The derivative of the cubic has found the foci of an ellipse determined by the roots of the original polynomial.

    Triangle formed by three complex roots with its Steiner inellipse tangent at the side midpoints. The two zeros of the derivative are marked at the two foci of the ellipse.
    Figure 3. Marden’s theorem: the two zeros w₁ and w₂ of the derivative are exactly the foci of the Steiner inellipse of the triangle formed by z₁, z₂, and z₃.

    A concrete example

    Let us see the theorem in action. Choose the three roots

    z1=−2, z2=1+2i, z3=1−2i.

    The corresponding cubic is

    p(z) = (z+2) (z−1−2i) (z−1+2i).

    Multiplying the conjugate factors first gives

    (z−1−2i) (z−1+2i) = z2 −2z +5.

    Hence

    p(z) = (z+2) ( z2 −2z +5 ) = z3 +z +10.

    Now differentiate:

    p′(z) = 3z2 +1.

    The critical points satisfy

    3z2 +1 =0,

    so

    z = ± i 3 .

    Marden’s theorem now tells us something geometric that would be very difficult to guess merely by looking at the polynomial:

    The two foci of the Steiner inellipse are

    i3 and − i3.
    Triangle with complex roots negative 2, 1 plus 2i, and 1 minus 2i, together with its Steiner inellipse. The two foci are located at i over square root of 3 and negative i over square root of 3.
    Figure 4. For p(z) = z³ + z + 10, the roots form the displayed triangle, while the zeros ±i/√3 of p′(z) are exactly the foci of its Steiner inellipse.

    Why is this so surprising?

    The derivative is usually introduced as an analytic object: it measures instantaneous rate of change, gives the slope of a tangent line, and locates critical points.

    Here it is doing something that looks completely different.

    Start with three complex numbers. Use them as the roots of a cubic. Differentiate the polynomial. Solve one quadratic equation. The two answers are not merely points somewhere inside the triangle—they are the two foci of a distinguished ellipse.

    In symbols, the chain of ideas is

    three roots → triangle → differentiate → two critical points → two foci.

    The same Steiner inellipse therefore admits two very different descriptions.

    In our earlier article, it appeared from an area condition: a point lies on the ellipse precisely when three corner triangles have total area equal to half the area of the original triangle.

    Here the ellipse appears from differentiation: its two foci are the zeros of the derivative of the cubic whose roots are the vertices of the triangle.

    This is a beautiful example of a recurring theme in mathematics: algebra, calculus, and geometry can encode the same object in completely different ways.

    References and further reading

    1. D. Kalman, “An Elementary Proof of Marden’s Theorem” , The American Mathematical Monthly, Vol. 115, No. 4 (2008), pp. 330–338.
    2. A. Eydelzon, “On a New Property of the Steiner Inellipse” , The American Mathematical Monthly, Vol. 127, No. 10 (2020), pp. 933–935.