Sr Examen
Descomponer -y^4-y^2-8 al cuadrado
Solución
Factorización
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/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ | 3/4 |atan\\/ 31 /| 3/4 |atan\\/ 31 /|| | 3/4 |atan\\/ 31 /| 3/4 |atan\\/ 31 /|| | 3/4 |atan\\/ 31 /| 3/4 |atan\\/ 31 /|| | 3/4 |atan\\/ 31 /| 3/4 |atan\\/ 31 /|| |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 //
$$\left(x + \left(2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}\right)\right) \left(x + \left(2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}\right)\right)$$
(((x + 2^(3/4)*sin(atan(sqrt(31))/2) + i*2^(3/4)*cos(atan(sqrt(31))/2))*(x + 2^(3/4)*sin(atan(sqrt(31))/2) - i*2^(3/4)*cos(atan(sqrt(31))/2)))*(x - 2^(3/4)*sin(atan(sqrt(31))/2) + i*2^(3/4)*cos(atan(sqrt(31))/2)))*(x - 2^(3/4)*sin(atan(sqrt(31))/2) - i*2^(3/4)*cos(atan(sqrt(31))/2))
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(- y^{4} - y^{2}\right) - 8$$
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
$$a = -1$$
$$b = -1$$
$$c = -8$$
Entonces
$$m = \frac{1}{2}$$
$$n = - \frac{31}{4}$$
Pues,
$$- \left(y^{2} + \frac{1}{2}\right)^{2} - \frac{31}{4}$$
$$\left(- y^{4} - y^{2}\right) - 8$$
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
$$a = -1$$
$$b = -1$$
$$c = -8$$
Entonces
$$m = \frac{1}{2}$$
$$n = - \frac{31}{4}$$
Pues,
$$- \left(y^{2} + \frac{1}{2}\right)^{2} - \frac{31}{4}$$
Unión de expresiones racionales
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2 / 2\ -8 + y *\-1 - y /
$$y^{2} \left(- y^{2} - 1\right) - 8$$
-8 + y^2*(-1 - y^2)