And also, the average of the roots of P is the average of the roots of P'. Not sure about the name of this theorem but it is also really cool.

Wow. This puts a new light on the cubic equations of state, where the derivatives are essential in finding the liquid and gas volume at an equilibrium.

This was very cool! I'm always amazed by how much interesting structure there is behind seemingly "simple" topics, like roots of a cubic and its derivative here.

So if you add a constant C to the P, the roots change of course, but not those of P'. The roots of P' will remain the foci of any ellipses formed by roots of P + C. Interesting.

Man, this is absolutely awesome from you. You don't stop surprising me.

And if you collapse this to a line (example only real roots), there must be a 0 of the derivative between each real root. Since each line segment between 2 roots is the convex hull. Nice.

Inellipse is ellipse inscribed within triangle. Personally I hope this term is consistent with ellipses inscribed in other polygons. Good to see you post again Dr. Peyam. We're all busy at work.

Quite interesting. Dankeschön. مرسی. 😊

amazing!! math is so incredible

Can you use the Vieta formulas for the roots (of the original and the derivative), to get a relationship between the roots of the original and the derivative? This might work since there is a simple closed form for the coefficients of the derivative of a polynomial.

New Dr Peyam upload :DD

Then for the cherry on top, the root of P’’ is center of the ellipse which is the midpoint between the roots of P’ (foci). Overall it’s a geometric jewel!

I'm not sure why but I find this amazing.

Very amazing

ha, the gauss-lucas proof was on my homework that was due last week! fun theorem

Thank you sir, I really appreciate your effort in trying to breakdown and explain the calculus topics. I don't know if U can do a video involving csc, cot and other variables in a math problem sir, thanks.

Peyam is back!

omg geometry my weakness prefer algebra but the Lucas hull container of lower differential form is beautiful & l was gobsmacked by the beautiful fact that the interior lower differential roots are foci of an ellipse....a beeeeeudiful property . Dr Peyam the best math guru KZfaq influencer . 🤪👍

I want a quintic example.

Great😀

7:42 🥁

Thx

NO WAY!!! Wie geil ist das denn!?

Peyam your magic is inspirational ....... makes me imagine.....seeing two set of planar triangulated roots for cubic polynomial and quadratic polynomials solutions.......whose first derivative and integral polynomial zero point solutions have group symmetries G1((1,2,3)^2)... and ....G2((4,5)^3)...meaning we can interpret....the differential equation integral(G2((4,5)^3)) = derivative(G1((1,2,3)^2)) has a 5 point closed symmetry ...Let's quick summarize..... we put together a pair of zero point calculus functions...and the math looks 'OK' OK.....let's imagine rotating both cubic and quadratic solution zero point roots, so that each root rotates 'pi/2' radians and then assume, all roots share the same angle of parallelism.... look ....a far hand is up........may we ask a physic's question? ......Are the five rotated roots the same as Lagrange's orbital set of 5 orbiting 'stationary' points derived from the symmetry generated by arithmetically combining a pair of calculus connected closed geometries with points mapped to.... or covered by....arithmetic subgroups G1 and G2 analytically extending to a super group G3((G1 * G2) + (G1+ G2))) line endpoint operands [...] calculated using arithmetic field point operations computed on a Gaussian (field point) clock calculator . The super group G3 virtually extends the cubic and quadratic polynomial zero point G1 and G2 solutions factored by both virtual(3D) and real(2D) vector spaces?

this reminds me of the Gershgorin circle theorem. is there a relation?

So the effect of varying the constant term of P would be to rotate the triangle, while maintaining contact whit the elipse?

Does the 2nd theorem generalizes in some way to any degree? Conjectures?

Woa 👌🔥

I would like to see this apart from i.

Too beautifil

Not that it affected the presentation much, but I did notice at the start that instead of saying "x^3 minus 4x^2 plus 6x minus 4", the additions and subtractions were transposed. Only a simple oops... and doesn't really detract as the error doesn't resurface. I do enjoy the enthusiasm for the subject!😀

So the Gauss Lucas Theorem would lead to each successive derivative having the convex hull of its roots being a subset of the convex of the previous derivative? Would the also apply for collinear sets of roots? E.g., if P(z)=0 has only real solutions, all derivatives of P(z) are guaranteed to only have real solutions?

You know me and my boy Gauss came up with this devious theorem together

Even higher order derivative roots.

Just to be clear, this works with complex coefficients aswell?

What happen with 4th or 5th degree case?

So what if the roots of P are all on the real line? What’s the convex hull then?

so if all 3 roots of the cubic are real then the roots of the derivative must also be real and between the largest and smallest roots of the cubic?

Thay x =1 thì f(1)= 1-4+6-4#0; x=2 thì f(2)=8-16+12-4=0; f(x)= (x-2).g(x)

Out of curiosity does this work for non complex roots? Or is that just nonsense

👍

Is this P geometric or algebraic? Because VLT dont like it.

I don't understand fully. ;-) But Good work.

Want to learn math till i can write from both hand actively 😄🤝

The théorème : Gauss Lucas

You coulda been a contender

i just noticed that this can only happen when the P ( x ) has only one real root.

The coefficients of P(x) look like the 4th row of Pascal's triangle [ 1 -4 6 -4 1 ]. Since x is not 0 so multiply x to get (x-1)^4 = 1. The 4 roots of unity are +/- 1 and +/- i => x = 2, 0, 1 +/- i (x is not 0 so discard that root). x = 2, 1+i and 1-i (No guessing of integer roots required).

Holly

500th Viewer

I L.O.V.E U

American long division LMAO