---
tag: A56086N8XQ
kind: statement
role: theorem
name: Gauss–Lucas theorem
msc: [26C10, 30C15, 12D10]
uses: [0Q9EZS90GY, 00EHX6FWC0, 0TCHAS04WK, 09FTAET5Z4]
about: [019MA1VTR2, 0FQSDC2EQ2]
sources:
  - url: https://en.wikipedia.org/wiki/Gauss–Lucas_theorem
same_as:
  wikidata: Q1008566
  wikipedia: Gauss–Lucas theorem
---

## Statement

Let $P$ be a nonconstant polynomial with coefficients in the [[complex numbers|09FTAET5Z4]], and let $P'$ be its derivative.
Then every [[root|00EHX6FWC0]] of $P'$ lies in the [[convex hull|0TCHAS04WK]] of the set of roots of $P$, viewed as points of the plane $\C$.

## Remarks

Proof idea: write $P(z) = c \prod_{i=1}^{n} (z - z_i)$, so that $\frac{P'(z)}{P(z)} = \sum_{i=1}^{n} \frac{1}{z - z_i}$ when $P(z) \ne 0$.
Let $w$ be a root of $P'$; if $P(w) = 0$ there is nothing to prove.
Otherwise $\sum_i \frac{1}{w - z_i} = 0$; taking complex conjugates gives $\sum_i \frac{w - z_i}{\abs{w - z_i}^2} = 0$.
So $w = \sum_i \lambda_i z_i$ with $\lambda_i = \abs{w - z_i}^{-2} \big/ \sum_j \abs{w - z_j}^{-2}$, a convex combination of the $z_i$.

Example: $P(z) = z^3 - 1$ has roots the three cube roots of unity, and $P'(z) = 3z^2$ has the root $0$, the centre of the triangle they form.

Take a real polynomial whose roots are all real.
Then the theorem says that the roots of $P'$ lie between the smallest and the largest root of $P$, which also follows from [[Rolle's theorem|0194DY35BG]].

## Explanations

### L1 human

Mark the roots of a polynomial as points in the plane and stretch a rubber band around them.
The roots of the derivative all lie inside the band.
Differentiating never moves the roots outside the region spanned by the original ones.

### L2 human

The theorem is a complex analogue of Rolle's theorem.
Applied repeatedly, it shows that the roots of all derivatives $P^{(k)}$, $k < \deg P$, lie in the convex hull of the roots of $P$.
It is a basic tool in the geometry of polynomials and is used, for example, in a short proof of [[Bernstein's inequality for polynomials|F0YGC89S9R]].