From Newsgroup: sci.math



----------------------------------

In french and in an american

---------------------------------



Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

Il s'agit d'une cubique poss|-dant trois racines (une r|-elle, deux complexes).

Le but n'est pas tant de d|-terminer ces trois racines, c'est tr|?s
facile.

Le but est de les repr|-senter toutes les trois sur un plan cart|-sien (ce
que l'Intelligence artificielle juge impossible), alors qu'un simple
|-l|?ve de lyc|-e, qui a compris le principe pour le faire, va pouvoir dessiner des diagrammes d'une particuli|?re beaut|- en utilisant DESMOS,
le logiciel conseill|- par le tr|?s excellent Python, et y placer
rapidement les racines de mani|?re visible et irr|-futable.

---------------------------------------------------------------------------------------------------

Today we're going to study the curve f(x)=x-|+6x-#+13x+10.

It's a cubic with three roots (one real, two complex).

The goal isn't so much to determine these three roots; that's very easy.

The goal is to represent all three on a Cartesian plane (something that artificial intelligence considers impossible), while a simple high school student, who understands the principle of doing so, will be able to draw particularly beautiful diagrams using DESMOS, the software recommended by
the excellent Python, and quickly place the roots in a visible and
irrefutable manner.


R.H.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Op 30-8-2025 om 15:27 schreef Richard Hachel:

----------------------------------

-aIn french and in an american

---------------------------------

Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

Il s'agit d'une cubique poss|-dant trois racines (une r|-elle, deux complexes).

Le but n'est pas tant de d|-terminer ces trois racines, c'est tr|?s facile.

Le but est de les repr|-senter toutes les trois sur un plan cart|-sien (ce que l'Intelligence artificielle juge impossible), alors qu'un simple
|-l|?ve de lyc|-e, qui a compris le principe pour le faire, va pouvoir dessiner des diagrammes d'une particuli|?re beaut|- en utilisant DESMOS,
le logiciel conseill|- par le tr|?s excellent Python, et y placer
rapidement les racines de mani|?re visible et irr|-futable.

---------------------------------------------------------------------------------------------------

Today we're going to study the curve f(x)=x-|+6x-#+13x+10.

It's a cubic with three roots (one real, two complex).

The goal isn't so much to determine these three roots; that's very easy.

The goal is to represent all three on a Cartesian plane (something that artificial intelligence considers impossible), while a simple high
school student, who understands the principle of doing so, will be able
to draw particularly beautiful diagrams using DESMOS, the software recommended by the excellent Python, and quickly place the roots in a visible and irrefutable manner.

R.H.

There is one root in the real numbers and two additional roots in the
complex numbers, as can be visualized with desmos 2d/3d:

https://www.desmos.com/calculator/i7lclqgt26
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Le 30/08/2025 |a 20:47, sobriquet a |-crit :

Op 30-8-2025 om 15:27 schreef Richard Hachel:

----------------------------------

-aIn french and in an american

---------------------------------



Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

Il s'agit d'une cubique poss|-dant trois racines (une r|-elle, deux
complexes).

Le but n'est pas tant de d|-terminer ces trois racines, c'est tr|?s facile. >>
Le but est de les repr|-senter toutes les trois sur un plan cart|-sien (ce >> que l'Intelligence artificielle juge impossible), alors qu'un simple
|-l|?ve de lyc|-e, qui a compris le principe pour le faire, va pouvoir
dessiner des diagrammes d'une particuli|?re beaut|- en utilisant DESMOS,
le logiciel conseill|- par le tr|?s excellent Python, et y placer
rapidement les racines de mani|?re visible et irr|-futable.


---------------------------------------------------------------------------------------------------

Today we're going to study the curve f(x)=x-|+6x-#+13x+10.

It's a cubic with three roots (one real, two complex).

The goal isn't so much to determine these three roots; that's very easy.

The goal is to represent all three on a Cartesian plane (something that
artificial intelligence considers impossible), while a simple high
school student, who understands the principle of doing so, will be able
to draw particularly beautiful diagrams using DESMOS, the software
recommended by the excellent Python, and quickly place the roots in a
visible and irrefutable manner.


R.H.

There is one root in the real numbers and two additional roots in the complex numbers, as can be visualized with desmos 2d/3d:

https://www.desmos.com/calculator/i7lclqgt26

Sauf que tes racines |a la con ne passent pas par y=0.

Et que toujours, toujours, toujours, de v|-ritables salopards, se vantant d'avoir des couilles comme des past|?ques, viendront ouvrir leur gueule
pour montrer qu'ils ont des cacahu|?tes plus grosses que les miennes.

Quelle belle bande de tar|-s, les hommes.

Vous voulez vraiment que je vous montre mes couilles, tas de crapauds?

LOL.

Vous allez chier dans vos frocs, ce qui est rigolo ; puis devenir
m|-chants, ce qui l'est moins.

L'histoire, je la connais, j'ai lu toute la litt|-rature
franco-britannique.

J'ai m|-me lu la science-fiction am|-ricaine.

Je ne suis pas de la derni|?re averse.

Pour revenir en charte : qu'est ce que c'est que vos racines |a la con, et
vos diagrammes d|-biles?

Mais vous |-tes de grands malades les mecs.

R.H.


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Le 30/08/2025 |a 20:47, sobriquet a |-crit :

Op 30-8-2025 om 15:27 schreef Richard Hachel:

----------------------------------

-aIn french and in an american

---------------------------------

https://www.desmos.com/calculator/i7lclqgt26

J'ai ri.

Qu'est ce que l'on peut faire d'autre, sinon rire?

R.H.


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Op 30-8-2025 om 21:33 schreef Richard Hachel:

Le 30/08/2025 |a 20:47, sobriquet a |-crit :

Op 30-8-2025 om 15:27 schreef Richard Hachel:

----------------------------------

-a-aIn french and in an american

---------------------------------



Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

Il s'agit d'une cubique poss|-dant trois racines (une r|-elle, deux
complexes).

Le but n'est pas tant de d|-terminer ces trois racines, c'est tr|?s
facile.

Le but est de les repr|-senter toutes les trois sur un plan cart|-sien
(ce que l'Intelligence artificielle juge impossible), alors qu'un
simple |-l|?ve de lyc|-e, qui a compris le principe pour le faire, va
pouvoir dessiner des diagrammes d'une particuli|?re beaut|- en
utilisant DESMOS, le logiciel conseill|- par le tr|?s excellent Python, >>> et y placer rapidement les racines de mani|?re visible et irr|-futable.


---------------------------------------------------------------------------------------------------

Today we're going to study the curve f(x)=x-|+6x-#+13x+10.

It's a cubic with three roots (one real, two complex).

The goal isn't so much to determine these three roots; that's very easy. >>>
The goal is to represent all three on a Cartesian plane (something
that artificial intelligence considers impossible), while a simple
high school student, who understands the principle of doing so, will
be able to draw particularly beautiful diagrams using DESMOS, the
software recommended by the excellent Python, and quickly place the
roots in a visible and irrefutable manner.


R.H.

There is one root in the real numbers and two additional roots in the
complex numbers, as can be visualized with desmos 2d/3d:

https://www.desmos.com/calculator/i7lclqgt26

Sauf que tes racines |a la con ne passent pas par y=0.

Et que toujours, toujours, toujours, de v|-ritables salopards, se vantant d'avoir des couilles comme des past|?ques, viendront ouvrir leur gueule
pour montrer qu'ils ont des cacahu|?tes plus grosses que les miennes.

Quelle belle bande de tar|-s, les hommes.

Vous voulez vraiment que je vous montre mes couilles, tas de crapauds?

LOL.

Vous allez chier dans vos frocs, ce qui est rigolo ; puis devenir
m|-chants, ce qui l'est moins.
L'histoire, je la connais, j'ai lu toute la litt|-rature franco-britannique.

J'ai m|-me lu la science-fiction am|-ricaine.

Je ne suis pas de la derni|?re averse.

Pour revenir en charte : qu'est ce que c'est que vos racines |a la con,
et vos diagrammes d|-biles?
Mais vous |-tes de grands malades les mecs.
R.H.

There is the Cartesian plane (x,y,0) where we can visualize the complex
roots (x+iy) such that f(x+iy)=0,
where f(x+iy)=(x+iy)-|+6(x+iy)-#+13(x+iy)+10


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Le 30/08/2025 |a 22:01, sobriquet a |-crit :

Op 30-8-2025 om 21:33 schreef Richard Hachel:

Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

https://www.desmos.com/calculator/i7lclqgt26

There is the Cartesian plane (x,y,0) where we can visualize the complex roots (x+iy) such that f(x+iy)=0,
where f(x+iy)=(x+iy)-|+6(x+iy)-#+13(x+iy)+10

Je ne vois aucune courbe, r|-elle ou imaginaire, passer par les deux
points indiqu|-s.

Qui d'ailleurs ne sont m|-me pas sur x'Ox.

R.H.




--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Op 30-8-2025 om 22:19 schreef Richard Hachel:

Le 30/08/2025 |a 22:01, sobriquet a |-crit :

Op 30-8-2025 om 21:33 schreef Richard Hachel:

Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

https://www.desmos.com/calculator/i7lclqgt26

There is the Cartesian plane (x,y,0) where we can visualize the
complex roots (x+iy) such that f(x+iy)=0,
where f(x+iy)=(x+iy)-|+6(x+iy)-#+13(x+iy)+10

Je ne vois aucune courbe, r|-elle ou imaginaire, passer par les deux
points indiqu|-s.

Qui d'ailleurs ne sont m|-me pas sur x'Ox.

R.H.

Here you can see the curve in red on the blue surface that represents
the real part of the function applied to complex numbers:

https://www.desmos.com/3d/pstad1rh6m


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Le 30/08/2025 |a 22:39, sobriquet a |-crit :

Here you can see the curve in red on the blue surface that represents
the real part of the function applied to complex numbers:

Tout cela tourne au plus profond ridicule.

Nan, laissez, laissez, je vais le faire.

R.H.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Op 30-8-2025 om 22:19 schreef Richard Hachel:

Le 30/08/2025 |a 22:01, sobriquet a |-crit :

Op 30-8-2025 om 21:33 schreef Richard Hachel:

Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

https://www.desmos.com/calculator/i7lclqgt26

There is the Cartesian plane (x,y,0) where we can visualize the
complex roots (x+iy) such that f(x+iy)=0,
where f(x+iy)=(x+iy)-|+6(x+iy)-#+13(x+iy)+10

Je ne vois aucune courbe, r|-elle ou imaginaire, passer par les deux
points indiqu|-s.

Qui d'ailleurs ne sont m|-me pas sur x'Ox.

R.H.

Here you can even drag the slider for the c variable to the left to
reveal the Cartesian plane containing the complex roots and the
associated curve (where the real part is -2) that is embedded in the
imaginary part of the function f applied to complex numbers.

https://i.imgur.com/AoWJ6bQ.png

https://www.desmos.com/3d/y6cfikctgb
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/30/2025 6:27 AM, Richard Hachel wrote:

----------------------------------

-aIn french and in an american

---------------------------------

Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

Il s'agit d'une cubique poss|-dant trois racines (une r|-elle, deux complexes).

Le but n'est pas tant de d|-terminer ces trois racines, c'est tr|?s facile.

Le but est de les repr|-senter toutes les trois sur un plan cart|-sien (ce que l'Intelligence artificielle juge impossible), alors qu'un simple
|-l|?ve de lyc|-e, qui a compris le principe pour le faire, va pouvoir dessiner des diagrammes d'une particuli|?re beaut|- en utilisant DESMOS,
le logiciel conseill|- par le tr|?s excellent Python, et y placer
rapidement les racines de mani|?re visible et irr|-futable.

---------------------------------------------------------------------------------------------------

Today we're going to study the curve f(x)=x-|+6x-#+13x+10.

It's a cubic with three roots (one real, two complex).

The goal isn't so much to determine these three roots; that's very easy.

The goal is to represent all three on a Cartesian plane (something that artificial intelligence considers impossible), while a simple high
school student, who understands the principle of doing so, will be able
to draw particularly beautiful diagrams using DESMOS, the software recommended by the excellent Python, and quickly place the roots in a visible and irrefutable manner.

Rendering a root as in (x, y)? Well, that is just plotting that point.
So, we plot all roots. Humm... If you like roots, perhaps, when you get
bored, try to implement my multijulia algorithm. Paul was kind enough to
do a little write up about it:

https://paulbourke.net/fractals/multijulia

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Op 30-8-2025 om 23:55 schreef Chris M. Thomasson:

On 8/30/2025 6:27 AM, Richard Hachel wrote:

----------------------------------

-a-aIn french and in an american

---------------------------------



Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

Il s'agit d'une cubique poss|-dant trois racines (une r|-elle, deux
complexes).

Le but n'est pas tant de d|-terminer ces trois racines, c'est tr|?s facile. >>
Le but est de les repr|-senter toutes les trois sur un plan cart|-sien
(ce que l'Intelligence artificielle juge impossible), alors qu'un
simple |-l|?ve de lyc|-e, qui a compris le principe pour le faire, va
pouvoir dessiner des diagrammes d'une particuli|?re beaut|- en utilisant
DESMOS, le logiciel conseill|- par le tr|?s excellent Python, et y
placer rapidement les racines de mani|?re visible et irr|-futable.

---------------------------------------------------------------------------------------------------

Today we're going to study the curve f(x)=x-|+6x-#+13x+10.

It's a cubic with three roots (one real, two complex).

The goal isn't so much to determine these three roots; that's very easy.

The goal is to represent all three on a Cartesian plane (something
that artificial intelligence considers impossible), while a simple
high school student, who understands the principle of doing so, will
be able to draw particularly beautiful diagrams using DESMOS, the
software recommended by the excellent Python, and quickly place the
roots in a visible and irrefutable manner.

Rendering a root as in (x, y)? Well, that is just plotting that point.
So, we plot all roots. Humm... If you like roots, perhaps, when you get bored, try to implement my multijulia algorithm. Paul was kind enough to
do a little write up about it:

https://paulbourke.net/fractals/multijulia

Plants also have roots.

https://www.desmos.com/calculator/vu0byi2iyt
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Le 30/08/2025 |a 22:39, sobriquet a |-crit :

Op 30-8-2025 om 22:19 schreef Richard Hachel:

Le 30/08/2025 |a 22:01, sobriquet a |-crit :

Op 30-8-2025 om 21:33 schreef Richard Hachel:

Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

https://www.desmos.com/calculator/i7lclqgt26

There is the Cartesian plane (x,y,0) where we can visualize the
complex roots (x+iy) such that f(x+iy)=0,
where f(x+iy)=(x+iy)-|+6(x+iy)-#+13(x+iy)+10

Je ne vois aucune courbe, r|-elle ou imaginaire, passer par les deux
points indiqu|-s.

Qui d'ailleurs ne sont m|-me pas sur x'Ox.

R.H.

Here you can see the curve in red on the blue surface that represents
the real part of the function applied to complex numbers:

https://www.desmos.com/3d/pstad1rh6m

Cartesian draw is better.

Real root in red.

Complex roots in blue

<http://nemoweb.net/jntp?D3F03tyNoFxDB1avIicVk_ybg1I@jntp/Data.Media:1>

R.H.
<https://www.nemoweb.net/?DataID=D3F03tyNoFxDB1avIicVk_ybg1I@jntp>
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/30/2025 2:58 PM, sobriquet wrote:

Op 30-8-2025 om 23:55 schreef Chris M. Thomasson:

On 8/30/2025 6:27 AM, Richard Hachel wrote:

----------------------------------

-a-aIn french and in an american

---------------------------------



Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

Il s'agit d'une cubique poss|-dant trois racines (une r|-elle, deux
complexes).

Le but n'est pas tant de d|-terminer ces trois racines, c'est tr|?s
facile.

Le but est de les repr|-senter toutes les trois sur un plan cart|-sien
(ce que l'Intelligence artificielle juge impossible), alors qu'un
simple |-l|?ve de lyc|-e, qui a compris le principe pour le faire, va
pouvoir dessiner des diagrammes d'une particuli|?re beaut|- en
utilisant DESMOS, le logiciel conseill|- par le tr|?s excellent Python, >>> et y placer rapidement les racines de mani|?re visible et irr|-futable.

---------------------------------------------------------------------------------------------------

Today we're going to study the curve f(x)=x-|+6x-#+13x+10.

It's a cubic with three roots (one real, two complex).

The goal isn't so much to determine these three roots; that's very easy. >>>
The goal is to represent all three on a Cartesian plane (something
that artificial intelligence considers impossible), while a simple
high school student, who understands the principle of doing so, will
be able to draw particularly beautiful diagrams using DESMOS, the
software recommended by the excellent Python, and quickly place the
roots in a visible and irrefutable manner.

Rendering a root as in (x, y)? Well, that is just plotting that point.
So, we plot all roots. Humm... If you like roots, perhaps, when you
get bored, try to implement my multijulia algorithm. Paul was kind
enough to do a little write up about it:

https://paulbourke.net/fractals/multijulia

Plants also have roots.

https://www.desmos.com/calculator/vu0byi2iyt

Nice fern! Love those IFS's.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Op 31-8-2025 om 02:31 schreef Richard Hachel:

Le 30/08/2025 |a 22:39, sobriquet a |-crit :

Op 30-8-2025 om 22:19 schreef Richard Hachel:

Le 30/08/2025 |a 22:01, sobriquet a |-crit :

Op 30-8-2025 om 21:33 schreef Richard Hachel:

Nous allons |-tudier aujourd'hui la courbe f(x)=x-|+6x-#+13x+10.

https://www.desmos.com/calculator/i7lclqgt26

There is the Cartesian plane (x,y,0) where we can visualize the
complex roots (x+iy) such that f(x+iy)=0,
where f(x+iy)=(x+iy)-|+6(x+iy)-#+13(x+iy)+10

Je ne vois aucune courbe, r|-elle ou imaginaire, passer par les deux
points indiqu|-s.

Qui d'ailleurs ne sont m|-me pas sur x'Ox.

R.H.

Here you can see the curve in red on the blue surface that represents
the real part of the function applied to complex numbers:

https://www.desmos.com/3d/pstad1rh6m

Cartesian draw is better.

Real root in red.

Complex roots in blue

<http://nemoweb.net/jntp?D3F03tyNoFxDB1avIicVk_ybg1I@jntp/Data.Media:1>

R.H. <https://www.nemoweb.net/?DataID=D3F03tyNoFxDB1avIicVk_ybg1I@jntp>

You come up with an unrelated 3rd degree polynomial
g(x)=-2x-|-12x-#-22x-12 that has nothing to do with the original
polynomial f(x)=x-|+6x-#+13x+10


https://www.desmos.com/calculator/3jwklmzeys


You're using crackpot math, where you put complex numbers with a
non-zero imaginary part on the real number line.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 31.08.2025 um 02:57 schrieb sobriquet:

You're using crackpot math [...]

You think? :-)

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Le 31/08/2025 |a 02:58, sobriquet a |-crit :

Op 31-8-2025 om 02:31 schreef Richard Hachel:

You come up with an unrelated 3rd degree polynomial
g(x)=-2x-|-12x-#-22x-12 that has nothing to do with the original
polynomial f(x)=x-|+6x-#+13x+10

No, YOU, you say that 3rd degree polynomial g(x)=-2x-|-12x-#-22x-12 has nothing to do with the original
polynomial f(x)=x-|+6x-#+13x+10

That's not what I'm saying.
I said that we could represent the pure or complex imaginary roots of all possible Cartesian functions on a simple Cartesian coordinate system Ox,
Oy.
This is what I'm doing here in the representation you created.
You've plotted the first function in red, and its imaginary twin function
in blue.
If you look closely at your own diagram, you'll see that the real root of
the first function appears on the red curve, and the two complex roots on
the blue curve (-2+i, -2-i).
Conversely, the two real roots of the blue curve appear, and its pure imaginary root is that of the red curve (i.e., x=2i).
This is a very elegant way of placing the complex or pure imaginary roots
of all Cartesian functions; you just need to determine each time which is
the associated twin function, the phenomenon being reciprocal, and you
will be able to invariably, and without error, place all the roots of the functions.

Yout draw is here :

<http://nemoweb.net/jntp?iJiOt9UbvLsyM2lpv-C_WRbAlEU@jntp/Data.Media:1>

R.H.
--
Ce message a |-t|- post|- avec Nemo : <https://www.nemoweb.net/?DataID=iJiOt9UbvLsyM2lpv-C_WRbAlEU@jntp>
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Le 31/08/2025 |a 04:18, Moebius a |-crit :

Am 31.08.2025 um 02:57 schrieb sobriquet:

You're using crackpot math [...]

You think? :-)

Je crois surtout avoir affaire |a une bien belle bande de cr|-tins.

Usenet se d|-grade d'ann|-es en ann|-es.

J'explique |o t|-te de con qu'il existe une possibilit|- |-l|-gante de
placer les racines d'une fonction quelconque, polynomiale, exponentielle, logarithmique, etc... sur un simple rep|?re cart|-sien, en utilisant l'axe
des x, mais de fa|oon invers|-e.

Le cr|-tin croit avoir affaire |a un troll.

Le nombre d'abrutis qui trainent sur les r|-seaux publics est affolant.

J'en arrive |a me demander si la t|-l|- n'est finalement pas mieux, en |-coutant les actualit|-s de monsieur Pujadas de la Roseraie-Macron, ou
les |-missions scientifiques des fr|?res Bogdanov nous expliquant les longueurs d'onde relativistes de l'horizon des trous noirs.

Bandes d'abrutis va.

Pauvres clowns.

R.H.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Op 31-8-2025 om 04:35 schreef Richard Hachel:

Le 31/08/2025 |a 02:58, sobriquet a |-crit :

Op 31-8-2025 om 02:31 schreef Richard Hachel:

You come up with an unrelated 3rd degree polynomial
g(x)=-2x-|-12x-#-22x-12 that has nothing to do with the original
polynomial f(x)=x-|+6x-#+13x+10

No, YOU, you say that 3rd degree polynomial g(x)=-2x-|-12x-#-22x-12 has nothing to do with the original polynomial f(x)=x-|+6x-#+13x+10

That's not what I'm saying.
I said that we could represent the pure or complex imaginary roots of
all possible Cartesian functions on a simple Cartesian coordinate system
Ox, Oy.
This is what I'm doing here in the representation you created.
You've plotted the first function in red, and its imaginary twin
function in blue.
If you look closely at your own diagram, you'll see that the real root
of the first function appears on the red curve, and the two complex
roots on the blue curve (-2+i, -2-i).
Conversely, the two real roots of the blue curve appear, and its pure imaginary root is that of the red curve (i.e., x=2i).
This is a very elegant way of placing the complex or pure imaginary
roots of all Cartesian functions; you just need to determine each time
which is the associated twin function, the phenomenon being reciprocal,
and you will be able to invariably, and without error, place all the
roots of the functions.

Yout draw is here :

<http://nemoweb.net/jntp?iJiOt9UbvLsyM2lpv-C_WRbAlEU@jntp/Data.Media:1>

R.H.

Nonsense, what I created was a curve that represents the intersection of
the plane x=-2 with the imaginary part of the function f applied to
complex numbers (the red curve as shown below, and it has nothing to do
with that function g you came up with (the green curve as shown below,
if we ignore the fact that we're plotting the curve in the plane x=-2).

https://www.desmos.com/3d/boamr5su9s

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From Newsgroup: sci.math

Le 31/08/2025 |a 05:55, sobriquet a |-crit :

Op 31-8-2025 om 04:35 schreef Richard Hachel:

No, YOU, you say that 3rd degree polynomial g(x)=-2x-|-12x-#-22x-12 has
nothing to do with the original polynomial f(x)=x-|+6x-#+13x+10

it has nothing to do with that function g you came up

Please don't talk nonsense, it hurts me, sniff...
Do you really think I pulled my function g(x) out of a hat?
No, not at all. Every real function f(x) has a twin imaginary function
g(x) where its pure or complex imaginary roots are found. The phenomenon
is also reciprocal. The complex roots of the other function are the real
roots of the first, and this is true for all universal functions.

R.H.




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From Newsgroup: sci.math

Le 31/08/2025 |a 06:13, Richard Hachel a |-crit-a:

Do you really think I pulled my function g(x) out of a hat?

out of your filthy ass
--
F.J.
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From Newsgroup: sci.math

Le 31/08/2025 |a 09:45, efji a |-crit :

Le 31/08/2025 |a 06:13, Richard Hachel a |-crit-a:

Do you really think I pulled my function g(x) out of a hat?

out of your filthy ass

Ce n'est pas tr|?s gentil.

Sniffff...

R.H.
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