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1+z+z^2+z^3+z^4+z^5+z^6+z^7+z^8+z^9+z^10+z^11=0 la ecuación

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Solución numérica:

Buscar la solución numérica en el intervalo [, ]

Solución

Ha introducido [src]
         2    3    4    5    6    7    8    9    10    11    
1 + z + z  + z  + z  + z  + z  + z  + z  + z  + z   + z   = 0
$$z^{11} + \left(z^{10} + \left(z^{9} + \left(z^{8} + \left(z^{7} + \left(z^{6} + \left(z^{5} + \left(z^{4} + \left(z^{3} + \left(z^{2} + \left(z + 1\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) = 0$$
Gráfica
Suma y producto de raíces [src]
suma
                       ___             ___           ___           ___           ___         ___     ___             ___
               1   I*\/ 3      1   I*\/ 3    1   I*\/ 3    1   I*\/ 3      I   \/ 3    I   \/ 3    \/ 3    I   I   \/ 3 
-1 - I + I + - - - ------- + - - + ------- + - - ------- + - + ------- + - - - ----- + - - ----- + ----- - - + - + -----
               2      2        2      2      2      2      2      2        2     2     2     2       2     2   2     2  
$$\left(\left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right) + \left(\left(\left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right) + \left(\left(\left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right) + \left(\left(\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) + \left(\left(-1 - i\right) + i\right)\right) + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right)\right) + \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right)\right)\right) + \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)$$
=
-1
$$-1$$
producto
        /          ___\ /          ___\ /        ___\ /        ___\ /        ___\ /      ___\ /  ___    \ /      ___\
        |  1   I*\/ 3 | |  1   I*\/ 3 | |1   I*\/ 3 | |1   I*\/ 3 | |  I   \/ 3 | |I   \/ 3 | |\/ 3    I| |I   \/ 3 |
-(-I)*I*|- - - -------|*|- - + -------|*|- - -------|*|- + -------|*|- - - -----|*|- - -----|*|----- - -|*|- + -----|
        \  2      2   / \  2      2   / \2      2   / \2      2   / \  2     2  / \2     2  / \  2     2/ \2     2  /
$$i \left(- \left(-1\right) i\right) \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \left(- \frac{\sqrt{3}}{2} - \frac{i}{2}\right) \left(- \frac{\sqrt{3}}{2} + \frac{i}{2}\right) \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right) \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)$$
=
-1
$$-1$$
-1
Respuesta rápida [src]
z1 = -1
$$z_{1} = -1$$
z2 = -I
$$z_{2} = - i$$
z3 = I
$$z_{3} = i$$
               ___
       1   I*\/ 3 
z4 = - - - -------
       2      2   
$$z_{4} = - \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
               ___
       1   I*\/ 3 
z5 = - - + -------
       2      2   
$$z_{5} = - \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
             ___
     1   I*\/ 3 
z6 = - - -------
     2      2   
$$z_{6} = \frac{1}{2} - \frac{\sqrt{3} i}{2}$$
             ___
     1   I*\/ 3 
z7 = - + -------
     2      2   
$$z_{7} = \frac{1}{2} + \frac{\sqrt{3} i}{2}$$
             ___
       I   \/ 3 
z8 = - - - -----
       2     2  
$$z_{8} = - \frac{\sqrt{3}}{2} - \frac{i}{2}$$
           ___
     I   \/ 3 
z9 = - - -----
     2     2  
$$z_{9} = - \frac{\sqrt{3}}{2} + \frac{i}{2}$$
        ___    
      \/ 3    I
z10 = ----- - -
        2     2
$$z_{10} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
            ___
      I   \/ 3 
z11 = - + -----
      2     2  
$$z_{11} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
z11 = sqrt(3)/2 + i/2
Respuesta numérica [src]
z1 = -1.0
z2 = -0.866025403784439 + 0.5*i
z3 = -0.5 + 0.866025403784439*i
z4 = 0.866025403784439 - 0.5*i
z5 = 0.5 + 0.866025403784439*i
z6 = 1.0*i
z7 = -1.0*i
z8 = -0.866025403784439 - 0.5*i
z9 = -0.5 - 0.866025403784439*i
z10 = 0.866025403784439 + 0.5*i
z11 = 0.5 - 0.866025403784439*i
z11 = 0.5 - 0.866025403784439*i