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¿Cómo vas a descomponer esta -(z^2+1)/(z-(-1-sqrt(3)*i)/2)^2+2*z/(z-(-1-sqrt(3)*i)/2) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src]
         2                            
      - z  - 1              2*z       
------------------- + ----------------
                  2              ___  
/           ___  \        -1 - \/ 3 *I
|    -1 - \/ 3 *I|    z - ------------
|z - ------------|             2      
\         2      /                    
$$\frac{2 z}{z - \frac{-1 - \sqrt{3} i}{2}} + \frac{- z^{2} - 1}{\left(z - \frac{-1 - \sqrt{3} i}{2}\right)^{2}}$$
(-z^2 - 1)/(z - (-1 - sqrt(3)*i)/2)^2 + (2*z)/(z - (-1 - sqrt(3)*i)/2)
Descomposición de una fracción [src]
1 - 2*(1 + i*sqrt(3))/(1 + 2*z + i*sqrt(3))^2
$$1 - \frac{2 \left(1 + \sqrt{3} i\right)}{\left(2 z + 1 + \sqrt{3} i\right)^{2}}$$
        /        ___\   
      2*\1 + I*\/ 3 /   
1 - --------------------
                       2
    /              ___\ 
    \1 + 2*z + I*\/ 3 / 
Simplificación general [src]
  /      2     /              ___\\
4*\-1 - z  + z*\1 + 2*z + I*\/ 3 //
-----------------------------------
                           2       
        /              ___\        
        \1 + 2*z + I*\/ 3 /        
$$\frac{4 \left(- z^{2} + z \left(2 z + 1 + \sqrt{3} i\right) - 1\right)}{\left(2 z + 1 + \sqrt{3} i\right)^{2}}$$
4*(-1 - z^2 + z*(1 + 2*z + i*sqrt(3)))/(1 + 2*z + i*sqrt(3))^2
Respuesta numérica [src]
(-1.0 - z^2)/(0.5 + z + 0.866025403784439*i)^2 + 2.0*z/(0.5 + z + 0.866025403784439*i)
(-1.0 - z^2)/(0.5 + z + 0.866025403784439*i)^2 + 2.0*z/(0.5 + z + 0.866025403784439*i)
Unión de expresiones racionales [src]
  /      2     /              ___\\
4*\-1 - z  + z*\1 + 2*z + I*\/ 3 //
-----------------------------------
                           2       
        /              ___\        
        \1 + 2*z + I*\/ 3 /        
$$\frac{4 \left(- z^{2} + z \left(2 z + 1 + \sqrt{3} i\right) - 1\right)}{\left(2 z + 1 + \sqrt{3} i\right)^{2}}$$
4*(-1 - z^2 + z*(1 + 2*z + i*sqrt(3)))/(1 + 2*z + i*sqrt(3))^2
Denominador racional [src]
                       3                          3                           3                           3                                3                          2                    3                                   3
    /              ___\        /              ___\       3 /              ___\       2 /              ___\          ___ /              ___\        /              ___\  /              ___\          ___  2 /              ___\ 
- 4*\1 + 2*z - I*\/ 3 /  - 8*z*\1 + 2*z - I*\/ 3 /  - 8*z *\1 + 2*z - I*\/ 3 /  - 4*z *\1 + 2*z - I*\/ 3 /  - 4*I*\/ 3 *\1 + 2*z - I*\/ 3 /  + 4*z*\1 + 2*z + I*\/ 3 / *\1 + 2*z - I*\/ 3 /  - 4*I*\/ 3 *z *\1 + 2*z - I*\/ 3 / 
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                       3                                                                                                        
                                                                                                       /             2\                                                                                                         
                                                                                                       \4 + 4*z + 4*z /                                                                                                         
$$\frac{- 8 z^{3} \left(2 z + 1 - \sqrt{3} i\right)^{3} - 4 z^{2} \left(2 z + 1 - \sqrt{3} i\right)^{3} - 4 \sqrt{3} i z^{2} \left(2 z + 1 - \sqrt{3} i\right)^{3} + 4 z \left(2 z + 1 - \sqrt{3} i\right)^{3} \left(2 z + 1 + \sqrt{3} i\right)^{2} - 8 z \left(2 z + 1 - \sqrt{3} i\right)^{3} - 4 \left(2 z + 1 - \sqrt{3} i\right)^{3} - 4 \sqrt{3} i \left(2 z + 1 - \sqrt{3} i\right)^{3}}{\left(4 z^{2} + 4 z + 4\right)^{3}}$$
(-4*(1 + 2*z - i*sqrt(3))^3 - 8*z*(1 + 2*z - i*sqrt(3))^3 - 8*z^3*(1 + 2*z - i*sqrt(3))^3 - 4*z^2*(1 + 2*z - i*sqrt(3))^3 - 4*i*sqrt(3)*(1 + 2*z - i*sqrt(3))^3 + 4*z*(1 + 2*z + i*sqrt(3))^2*(1 + 2*z - i*sqrt(3))^3 - 4*i*sqrt(3)*z^2*(1 + 2*z - i*sqrt(3))^3)/(4 + 4*z + 4*z^2)^3
Denominador común [src]
                          ___              
                  1 + I*\/ 3               
1 - ---------------------------------------
                  2       ___           ___
    -1 + 2*z + 2*z  + I*\/ 3  + 2*I*z*\/ 3 
$$1 - \frac{1 + \sqrt{3} i}{2 z^{2} + 2 z + 2 \sqrt{3} i z - 1 + \sqrt{3} i}$$
1 - (1 + i*sqrt(3))/(-1 + 2*z + 2*z^2 + i*sqrt(3) + 2*i*z*sqrt(3))
Compilar la expresión [src]
           2                        
     -1 - z                2*z      
------------------ + ---------------
                 2               ___
/            ___\    1       I*\/ 3 
|1       I*\/ 3 |    - + z + -------
|- + z + -------|    2          2   
\2          2   /                   
$$\frac{2 z}{z + \frac{1}{2} + \frac{\sqrt{3} i}{2}} + \frac{- z^{2} - 1}{\left(z + \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{2}}$$
(-1 - z^2)/(1/2 + z + i*sqrt(3)/2)^2 + 2*z/(1/2 + z + i*sqrt(3)/2)
Parte trigonométrica [src]
           2                        
     -1 - z                2*z      
------------------ + ---------------
                 2               ___
/            ___\    1       I*\/ 3 
|1       I*\/ 3 |    - + z + -------
|- + z + -------|    2          2   
\2          2   /                   
$$\frac{2 z}{z + \frac{1}{2} + \frac{\sqrt{3} i}{2}} + \frac{- z^{2} - 1}{\left(z + \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{2}}$$
(-1 - z^2)/(1/2 + z + i*sqrt(3)/2)^2 + 2*z/(1/2 + z + i*sqrt(3)/2)
Combinatoria [src]
  /          2         ___\
4*\-1 + z + z  + I*z*\/ 3 /
---------------------------
                       2   
    /              ___\    
    \1 + 2*z + I*\/ 3 /    
$$\frac{4 \left(z^{2} + z + \sqrt{3} i z - 1\right)}{\left(2 z + 1 + \sqrt{3} i\right)^{2}}$$
4*(-1 + z + z^2 + i*z*sqrt(3))/(1 + 2*z + i*sqrt(3))^2
Potencias [src]
           2                        
     -1 - z                2*z      
------------------ + ---------------
                 2               ___
/            ___\    1       I*\/ 3 
|1       I*\/ 3 |    - + z + -------
|- + z + -------|    2          2   
\2          2   /                   
$$\frac{2 z}{z + \frac{1}{2} + \frac{\sqrt{3} i}{2}} + \frac{- z^{2} - 1}{\left(z + \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{2}}$$
(-1 - z^2)/(1/2 + z + i*sqrt(3)/2)^2 + 2*z/(1/2 + z + i*sqrt(3)/2)