In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, a branch of mathematics, the Gauss–Lucas theorem gives a
geometric relation between the
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
s of a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
''P'' and the roots of its
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
''P′''. The set of roots of a real or complex polynomial is a set of
points in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. The theorem states that the roots of ''P′'' all lie within the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of the roots of ''P'', that is the smallest
convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
containing the roots of ''P''. When ''P'' has a single root then this convex hull is a single point and when the roots lie on a
line then the convex hull is a
segment of this line. The Gauss–Lucas theorem, named after
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
and Félix Lucas, is similar in spirit to
Rolle's theorem.
Formal statement
If ''P'' is a (nonconstant) polynomial with complex coefficients, all
zeros of ''P′'' belong to the convex hull of the set of zeros of ''P''.
Special cases
It is easy to see that if ''P''(''x'') = ''ax''
2 + ''bx'' + ''c'' is a
second degree polynomial, the zero of ''P′''(''x'') = 2''ax'' + ''b'' is the
average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
of the roots of ''P''. In that case, the convex hull is the line segment with the two roots as endpoints and it is clear that the average of the roots is the middle point of the segment.
For a third degree complex polynomial ''P'' (
cubic function
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
) with three distinct zeros,
Marden's theorem
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 deriva ...
states that the zeros of ''P′'' are the foci of the
Steiner inellipse
In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse i ...
which is the unique ellipse tangent to the midpoints of the triangle formed by the zeros of ''P''.
For a fourth degree complex polynomial ''P'' (
quartic function
In algebra, a quartic function is a function of the form
:f(x)=ax^4+bx^3+cx^2+dx+e,
where ''a'' is nonzero,
which is defined by a polynomial of degree four, called a quartic polynomial.
A '' quartic equation'', or equation of the fourth de ...
) with four distinct zeros forming a concave
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
, one of the zeros of ''P'' lies within the convex hull of the other three; all three zeros of ''P′'' lie in two of the three triangles formed by the interior zero of ''P'' and two others zeros of ''P''.
In addition, if a polynomial of degree ''n'' of
real coefficients has ''n'' distinct real zeros