Simplificación general
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$$- y^{4} - 4 y^{2} - 6$$
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
| | |\/ 2 || | |\/ 2 ||| | | |\/ 2 || | |\/ 2 ||| | | |\/ 2 || | |\/ 2 ||| | | |\/ 2 || | |\/ 2 |||
| |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----|||
| 4 ___ | \ 2 /| 4 ___ | \ 2 /|| | 4 ___ | \ 2 /| 4 ___ | \ 2 /|| | 4 ___ | \ 2 /| 4 ___ | \ 2 /|| | 4 ___ | \ 2 /| 4 ___ | \ 2 /||
|x + \/ 6 *sin|-----------| + I*\/ 6 *cos|-----------||*|x + \/ 6 *sin|-----------| - I*\/ 6 *cos|-----------||*|x + - \/ 6 *sin|-----------| + I*\/ 6 *cos|-----------||*|x + - \/ 6 *sin|-----------| - I*\/ 6 *cos|-----------||
\ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(x + \left(\sqrt[4]{6} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} \right)} - \sqrt[4]{6} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{6} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} \right)} + \sqrt[4]{6} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{6} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} \right)} + \sqrt[4]{6} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{6} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} \right)} - \sqrt[4]{6} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} \right)}\right)\right)$$
(((x + 6^(1/4)*sin(atan(sqrt(2)/2)/2) + i*6^(1/4)*cos(atan(sqrt(2)/2)/2))*(x + 6^(1/4)*sin(atan(sqrt(2)/2)/2) - i*6^(1/4)*cos(atan(sqrt(2)/2)/2)))*(x - 6^(1/4)*sin(atan(sqrt(2)/2)/2) + i*6^(1/4)*cos(atan(sqrt(2)/2)/2)))*(x - 6^(1/4)*sin(atan(sqrt(2)/2)/2) - i*6^(1/4)*cos(atan(sqrt(2)/2)/2))
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(- y^{4} - 4 y^{2}\right) - 6$$
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
$$a = -1$$
$$b = -4$$
$$c = -6$$
Entonces
$$m = 2$$
$$n = -2$$
Pues,
$$- \left(y^{2} + 2\right)^{2} - 2$$
$$- y^{4} - 4 y^{2} - 6$$
$$- y^{4} - 4 y^{2} - 6$$
Parte trigonométrica
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$$- y^{4} - 4 y^{2} - 6$$
Compilar la expresión
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$$- y^{4} - 4 y^{2} - 6$$
Denominador racional
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$$- y^{4} - 4 y^{2} - 6$$
Unión de expresiones racionales
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$$y^{2} \left(- y^{2} - 4\right) - 6$$
$$- y^{4} - 4 y^{2} - 6$$