### Gauss-Lucas theorem

In complex analysis, a branch of mathematics, the **Gauss–Lucas theorem** gives a geometrical relation between the roots of a polynomial *P* and the roots of its derivative *P'*. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of *P'* all lie within the convex hull of the roots of *P*, that is the smallest convex polygon 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 and Félix Lucas is similar in spirit to Rolle's theorem.

## Contents

## 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*.^{[1]}

## 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 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.

In addition, if a polynomial of degree *n* of real coefficients has *n* distinct real zeros $x\_1\backslash cdots\backslash ,\; math>,\; we\; see,\; usingRolle\text{'}s\; theorem,\; that\; the\; zeros\; of\; the\; derivative\; polynomial\; are\; in\; the\; interval$ [x\_1,x\_n]\backslash ,$which\; is\; the\; convex\; hull\; of\; the\; set\; of\; roots.$

The convex hull of the roots of the polynomial $p\_n\; x^n+p\_\{n-1\}x^\{n-1\}+\backslash cdots\; p\_0$ particularly includes the point $-\backslash frac\{p\_\{n-1\}\}\{n\backslash cdot\; p\_n\}$.

## Proof

Over the complex numbers, *P* is a product of prime factors

- $P(z)=\; \backslash alpha\; \backslash prod\_\{i=1\}^n\; (z-a\_i)$

where the complex numbers $a\_1,\; a\_2,\; \backslash ldots,\; a\_n$ are the – not necessary distinct – zeros of the polynomial P, the complex number $\backslash alpha$ is the leading coefficient of P and n is the degree of P. Let z be any complex number for which $P(z)\; \backslash neq\; 0$. Then we have for the logarithmic derivative

- $\backslash frac\{P^\backslash prime(z)\}\{P(z)\}=\; \backslash sum\_\{i=1\}^n\; \backslash frac\{1\}\{z-a\_i\}.$

In particular, if z is a zero of $P^\backslash prime$ and still $P(z)\; \backslash neq\; 0$, then

- $\backslash sum\_\{i=1\}^n\; \backslash frac\{1\}\{z-a\_i\}=0\backslash $

or

- $\backslash \; \backslash sum\_\{i=1\}^n\; \backslash frac\{\backslash overline\{z\}-\backslash overline\{a\_i\}\; \}\; \{\backslash vert\; z-a\_i\backslash vert^2\}=0.$

This may also be written as

- $\backslash left(\backslash sum\_\{i=1\}^n\; \backslash frac\{1\}\{\backslash vert\; z-a\_i\backslash vert^2\}\backslash right)\backslash overline\{z\}=$

\sum_{i=1}^n \frac{1}{\vert z-a_i\vert^2}\overline{a_i}.

Taking their conjugates, we see that $z$ is a weighted sum with positive coefficients that sum to one, or the barycenter on affine coordinates, of the complex numbers $a\_i$ (with different mass assigned on each root whose weights collectively sum to 1).

If $P(z)=P^\backslash prime(z)=0$, then $z=1\backslash cdot\; z+0\backslash cdot\; a\_i$, and is still a convex combination of the roots of $P$.

## See also

- Marden's theorem
- Bôcher's theorem
- Sendov's conjecture
- Rational root theorem
- Routh–Hurwitz theorem
- Hurwitz's theorem (complex analysis)
- Descartes' rule of signs
- Rouché's theorem
- Sturm's theorem
- Properties of polynomial roots
- Gauss's lemma (polynomial)
- Polynomial function theorems for zeros
- Content (algebra)

## Notes

## References

- Morris Marden,
*Geometry of Polynomials*, AMS, 1966.

## External links

- Wolfram Demonstrations Project.