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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
 4    2    
x  + x  + 2
$$\left(x^{4} + x^{2}\right) + 2$$ 
x^4 + x^2 + 2
Simplificación general [src] 
     2    4
2 + x  + x
$$x^{4} + x^{2} + 2$$ 
2 + x^2 + x^4
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(x^{4} + x^{2}\right) + 2$$
Para eso usemos la fórmula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = 1$$
$$b = 1$$
$$c = 2$$
Entonces
$$m = \frac{1}{2}$$
$$n = \frac{7}{4}$$
Pues,
$$\left(x^{2} + \frac{1}{2}\right)^{2} + \frac{7}{4}$$
Factorización [src] 
/             /    /  ___\\              /    /  ___\\\ /             /    /  ___\\              /    /  ___\\\ /               /    /  ___\\              /    /  ___\\\ /               /    /  ___\\              /    /  ___\\\
|    4 ___    |atan\\/ 7 /|     4 ___    |atan\\/ 7 /|| |    4 ___    |atan\\/ 7 /|     4 ___    |atan\\/ 7 /|| |      4 ___    |atan\\/ 7 /|     4 ___    |atan\\/ 7 /|| |      4 ___    |atan\\/ 7 /|     4 ___    |atan\\/ 7 /||
|x + \/ 2 *sin|-----------| + I*\/ 2 *cos|-----------||*|x + \/ 2 *sin|-----------| - I*\/ 2 *cos|-----------||*|x + - \/ 2 *sin|-----------| + I*\/ 2 *cos|-----------||*|x + - \/ 2 *sin|-----------| - I*\/ 2 *cos|-----------||
\             \     2     /              \     2     // \             \     2     /              \     2     // \               \     2     /              \     2     // \               \     2     /              \     2     //
$$\left(x + \left(\sqrt[4]{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - \sqrt[4]{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + \sqrt[4]{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + \sqrt[4]{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - \sqrt[4]{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right)$$ 
(((x + 2^(1/4)*sin(atan(sqrt(7))/2) + i*2^(1/4)*cos(atan(sqrt(7))/2))*(x + 2^(1/4)*sin(atan(sqrt(7))/2) - i*2^(1/4)*cos(atan(sqrt(7))/2)))*(x - 2^(1/4)*sin(atan(sqrt(7))/2) + i*2^(1/4)*cos(atan(sqrt(7))/2)))*(x - 2^(1/4)*sin(atan(sqrt(7))/2) - i*2^(1/4)*cos(atan(sqrt(7))/2))
Combinatoria [src] 
     2    4
2 + x  + x
$$x^{4} + x^{2} + 2$$ 
2 + x^2 + x^4
Potencias [src] 
     2    4
2 + x  + x
$$x^{4} + x^{2} + 2$$ 
2 + x^2 + x^4
Denominador común [src] 
     2    4
2 + x  + x
$$x^{4} + x^{2} + 2$$ 
2 + x^2 + x^4
Respuesta numérica [src] 
2.0 + x^2 + x^4
2.0 + x^2 + x^4
Denominador racional [src] 
     2    4
2 + x  + x
$$x^{4} + x^{2} + 2$$ 
2 + x^2 + x^4
Compilar la expresión [src] 
     2    4
2 + x  + x
$$x^{4} + x^{2} + 2$$ 
2 + x^2 + x^4
Parte trigonométrica [src] 
     2    4
2 + x  + x
$$x^{4} + x^{2} + 2$$ 
2 + x^2 + x^4
Unión de expresiones racionales [src] 
     2 /     2\
2 + x *\1 + x /
$$x^{2} \left(x^{2} + 1\right) + 2$$ 
2 + x^2*(1 + x^2)
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