Gauss–Lucas theorem

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.

Going deeper

For a student

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.