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Simplificación de expresiones/ Descomposición de multiplicadores/ x^4+x^3+x^2+x+1

Factorizar el polinomio x^4+x^3+x^2+x+1

Expresión a simplificar:

Solución

Ha introducido [src] 
 4    3    2        
x  + x  + x  + x + 1
$$\left(x + \left(x^{2} + \left(x^{4} + x^{3}\right)\right)\right) + 1$$ 
x^4 + x^3 + x^2 + x + 1
Simplificación general [src] 
         2    3    4
1 + x + x  + x  + x
$$x^{4} + x^{3} + x^{2} + x + 1$$ 
1 + x + x^2 + x^3 + x^4
Factorización [src] 
/                       ___________\ /                       ___________\ /                       ___________\ /                       ___________\
|          ___         /       ___ | |          ___         /       ___ | |          ___         /       ___ | |          ___         /       ___ |
|    1   \/ 5         /  5   \/ 5  | |    1   \/ 5         /  5   \/ 5  | |    1   \/ 5         /  5   \/ 5  | |    1   \/ 5         /  5   \/ 5  |
|x + - - ----- + I*  /   - + ----- |*|x + - - ----- - I*  /   - + ----- |*|x + - + ----- + I*  /   - - ----- |*|x + - + ----- - I*  /   - - ----- |
\    4     4       \/    8     8   / \    4     4       \/    8     8   / \    4     4       \/    8     8   / \    4     4       \/    8     8   /
$$\left(x + \left(- \frac{\sqrt{5}}{4} + \frac{1}{4} - i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) \left(x + \left(- \frac{\sqrt{5}}{4} + \frac{1}{4} + i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}\right)\right) \left(x + \left(\frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right) \left(x + \left(\frac{1}{4} + \frac{\sqrt{5}}{4} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right)$$ 
(((x + 1/4 - sqrt(5)/4 + i*sqrt(5/8 + sqrt(5)/8))*(x + 1/4 - sqrt(5)/4 - i*sqrt(5/8 + sqrt(5)/8)))*(x + 1/4 + sqrt(5)/4 + i*sqrt(5/8 - sqrt(5)/8)))*(x + 1/4 + sqrt(5)/4 - i*sqrt(5/8 - sqrt(5)/8))
Respuesta numérica [src] 
1.0 + x + x^2 + x^3 + x^4
1.0 + x + x^2 + x^3 + x^4
Combinatoria [src] 
         2    3    4
1 + x + x  + x  + x
$$x^{4} + x^{3} + x^{2} + x + 1$$ 
1 + x + x^2 + x^3 + x^4
Parte trigonométrica [src] 
         2    3    4
1 + x + x  + x  + x
$$x^{4} + x^{3} + x^{2} + x + 1$$ 
1 + x + x^2 + x^3 + x^4
Compilar la expresión [src] 
         2    3    4
1 + x + x  + x  + x
$$x^{4} + x^{3} + x^{2} + x + 1$$ 
1 + x + x^2 + x^3 + x^4
Denominador racional [src] 
         2    3    4
1 + x + x  + x  + x
$$x^{4} + x^{3} + x^{2} + x + 1$$ 
1 + x + x^2 + x^3 + x^4
Denominador común [src] 
         2    3    4
1 + x + x  + x  + x
$$x^{4} + x^{3} + x^{2} + x + 1$$ 
1 + x + x^2 + x^3 + x^4
Unión de expresiones racionales [src] 
1 + x*(1 + x*(1 + x*(1 + x)))
$$x \left(x \left(x \left(x + 1\right) + 1\right) + 1\right) + 1$$ 
1 + x*(1 + x*(1 + x*(1 + x)))
Potencias [src] 
         2    3    4
1 + x + x  + x  + x
$$x^{4} + x^{3} + x^{2} + x + 1$$ 
1 + x + x^2 + x^3 + x^4
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