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Descomponer y^4+y^2+3 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src]
 4    2    
y  + y  + 3
$$\left(y^{4} + y^{2}\right) + 3$$
y^4 + y^2 + 3
Factorización [src]
/             /    /  ____\\              /    /  ____\\\ /             /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\
|    4 ___    |atan\\/ 11 /|     4 ___    |atan\\/ 11 /|| |    4 ___    |atan\\/ 11 /|     4 ___    |atan\\/ 11 /|| |      4 ___    |atan\\/ 11 /|     4 ___    |atan\\/ 11 /|| |      4 ___    |atan\\/ 11 /|     4 ___    |atan\\/ 11 /||
|x + \/ 3 *sin|------------| + I*\/ 3 *cos|------------||*|x + \/ 3 *sin|------------| - I*\/ 3 *cos|------------||*|x + - \/ 3 *sin|------------| + I*\/ 3 *cos|------------||*|x + - \/ 3 *sin|------------| - I*\/ 3 *cos|------------||
\             \     2      /              \     2      // \             \     2      /              \     2      // \               \     2      /              \     2      // \               \     2      /              \     2      //
$$\left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{11} \right)}}{2} \right)}\right)\right)$$
(((x + 3^(1/4)*sin(atan(sqrt(11))/2) + i*3^(1/4)*cos(atan(sqrt(11))/2))*(x + 3^(1/4)*sin(atan(sqrt(11))/2) - i*3^(1/4)*cos(atan(sqrt(11))/2)))*(x - 3^(1/4)*sin(atan(sqrt(11))/2) + i*3^(1/4)*cos(atan(sqrt(11))/2)))*(x - 3^(1/4)*sin(atan(sqrt(11))/2) - i*3^(1/4)*cos(atan(sqrt(11))/2))
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(y^{4} + y^{2}\right) + 3$$
Para eso usemos la fórmula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = 1$$
$$b = 1$$
$$c = 3$$
Entonces
$$m = \frac{1}{2}$$
$$n = \frac{11}{4}$$
Pues,
$$\left(y^{2} + \frac{1}{2}\right)^{2} + \frac{11}{4}$$
Simplificación general [src]
     2    4
3 + y  + y 
$$y^{4} + y^{2} + 3$$
3 + y^2 + y^4
Denominador racional [src]
     2    4
3 + y  + y 
$$y^{4} + y^{2} + 3$$
3 + y^2 + y^4
Respuesta numérica [src]
3.0 + y^2 + y^4
3.0 + y^2 + y^4
Compilar la expresión [src]
     2    4
3 + y  + y 
$$y^{4} + y^{2} + 3$$
3 + y^2 + y^4
Potencias [src]
     2    4
3 + y  + y 
$$y^{4} + y^{2} + 3$$
3 + y^2 + y^4
Parte trigonométrica [src]
     2    4
3 + y  + y 
$$y^{4} + y^{2} + 3$$
3 + y^2 + y^4
Combinatoria [src]
     2    4
3 + y  + y 
$$y^{4} + y^{2} + 3$$
3 + y^2 + y^4
Unión de expresiones racionales [src]
     2 /     2\
3 + y *\1 + y /
$$y^{2} \left(y^{2} + 1\right) + 3$$
3 + y^2*(1 + y^2)
Denominador común [src]
     2    4
3 + y  + y 
$$y^{4} + y^{2} + 3$$
3 + y^2 + y^4