How do the roots of a polynomial relate to the roots of its derivative? - Complex Analysis

This video presents a geometric proof of the Gauss-Lucas theorem, the fact that the roots of the derivative of a nonconstant polynomial sit within the convex hull of the roots of the polynomial.

I learned this argument from

Arnaud Chéritat, Yan Gao, Yafei Ou, Lei Tan. A refinement of the Gauss-Lucas theorem (after W. P. Thurston) . 2015. ffhal-01157602f

#complex #analysis #polynomial #phaseplot

[TheMathJuggler]

Insightful yet simple explanation of a beautiful theorem!

[andy_hay]

You were my favorite You made linear algebra super interesting

[yue-dongchen]

Thanks Professor for this beautifully animated introduction to this theorem. I'm revising for my Year 11 exams here in Australia and wondered about the name of the underlying theorem used in a taught technique to identify a repeated root of a polynomial, given its multiplicity — differentiate the polynomial down to a quadratic trinomial, and then test each solution in the original polynomial.

I'm sure I will come back to this video once I learn to work in the complex plane. But until then at least it's nice to know the theorem name even though I can't grasp it's geometric meaning.

[funfunny]

Professor I saw your vedio first time it awesome for me.Your explanation is so simple and easy.

[harinathan]

Truly beautiful explanation and proof.

[Roman-dtgz]

What a beautiful theorem. Thank you for the video!

[computage]

Hi! Thank you for the proof! It is very exciting! I have a question: can I somehow estimate positions of roots of polynomial derivative given only its roots (I mean not involving direct differentiation of the polynomial)? It looks like there roots of derivatives are some kind of barycentre of original roots, but i'm not sure.

[black_jack_meghav]

Love your videos as always, professor!

[aBigBadWolf]

Distractive pronunciation of root. Last time I checked, "root" is spoken as in "rooting for someone". Not sure the dramatisation at points is necessary. Good animations though. I liked the phase plot, is there a previous video where you go into more detail regarding the phase plot? Somehow the steps where you start talking about wiggling f(w) and introducing the gradient 04:45 were confusing to me. Not sure I understood your logic. Have to watch again later.

[Jop_pop]

I can't tell if I love or hate the way you say roots

Update: I hate it. Great video though!