polynomial with complex coefficients
The fundamental theorem of algebra - a visual proof
http://weitz.de/fund/
Der Fundamentalsatz der Algebra (anschaulicher Beweis)
Do not set ai=0, use checkboxes to switch on/off using coefficient ai.
The function f(x) CAS (1) has probably to be set manually if cell do not actualize - mark cell + Num Eval.
Hints:
ai complex coefficient of p(z) - input of values
polarcoordinates (mi; ωI)
z=(a ; φ) polarcoordinates - sliders on top - (0<=a<=10)
Zi = ai zi polynom summand in polarcoordinates =( mi ai ; ωi + i φ) -> vectors vi
c(t) parametric curve of polynom p(z) - graph p(z)
Pz = p(z)
App - Darstellung der 3. Nullstelle p(z): :
![Toolbar Image](/images/ggb/toolbar/mode_viewinfrontof.png)
polynomial with complex coefficients fkt
Find roots p(z) - z=ae^(φi)
The summands of the polynomial build a vector chain (blue). A root can be found if the vector chain of the coefficients is closed - returning to the origin.
![](data:image/png;base64,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)
- increase modulus (slider a) so far to have a complete free circular disc around the point of origin Z
- decrease modulus to intersect graph p(z) and origin Z=(0,0) - close vector chain
- zoom graph to position Pz (slider φ) as close as possible to the origin
- (a ; φ ) root in polarcoordinates ==> cartesian coordinates
- check calculated roots in Lcpx - next intersection
3D Visualisation p(z)=g(x,y) Surface (open view 3D Graphics in app)
![3D Visualisation p(z)=g(x,y) Surface (open view 3D Graphics in app)](https://beta.geogebra.org/resource/urp8yfas/Zkxm0pkgzPzbpRon/material-urp8yfas.png)
double root (real part ± imaginary part) - multiple point or repeated root
![double root (real part ± imaginary part) - multiple point or repeated root](https://beta.geogebra.org/resource/kvshqqm5/mDRwvsnCPIiRxPz4/material-kvshqqm5.png)