Marden's theorem
Encyclopedia
In mathematics
, Marden's theorem, named after Morris Marden, gives a geometric relationship between the zeroes of a third-degree polynomial
with complex
coefficients and the zeroes of its derivative
.
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:
By the Gauss–Lucas theorem, the root of the double derivative 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 ) the inscribed ellipse degenerates to a circle, and the derivative of has a double root at the center of the circle. Conversely, if the derivative has a double root, then the triangle must be equilateral .
A more general version of the theorem, due to , applies to polynomials whose degree may be higher than three, but that have only three roots , , and . 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 , , and .
attributes what is now known as Marden's theorem to 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Marden's theorem, named after Morris Marden, gives a geometric relationship between the zeroes of a third-degree polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
with complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
coefficients and the zeroes of its derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
.
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 , , and of a third-degree polynomial are non-collinear. There is a unique ellipse inscribed in the triangleTriangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
with vertices , , and tangentTangentIn geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
to the sides at their midpointMidpointThe midpoint is the middle point of a line segment. It is equidistant from both endpoints.-Formulas:...
s: the Steiner inellipseSteiner inellipseIn geometry, the Steiner inellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It is an example of an inconic. By comparison the inscribed circle of a triangle is another inconic that is tangent to the sides, but not necessarily at the...
. The fociFocus (geometry)In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...
of that ellipse are the zeroes of the derivative .
By the Gauss–Lucas theorem, the root of the double derivative must be the average of the two foci, which is the center point of the ellipse and the centroid
Centroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
of the triangle.
In the special case that the triangle is equilateral (as happens, for instance, for the polynomial ) the inscribed ellipse degenerates to a circle, and the derivative of has a double root at the center of the circle. Conversely, if the derivative has a double root, then the triangle must be equilateral .
A more general version of the theorem, due to , applies to polynomials whose degree may be higher than three, but that have only three roots , , and . 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 , , and .
attributes what is now known as Marden's theorem to 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
Mathematical Association of America
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
for his 2008 paper in the American Mathematical Monthly
American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....
describing the theorem.