Sr Examen
Descomponer -y^4+y^2-7 al cuadrado
Solución
Factorización
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$$\left(x + \left(\sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} - \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} + \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} + \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} - \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right)$$
(((x + 7^(1/4)*cos(atan(3*sqrt(3))/2) + i*7^(1/4)*sin(atan(3*sqrt(3))/2))*(x + 7^(1/4)*cos(atan(3*sqrt(3))/2) - i*7^(1/4)*sin(atan(3*sqrt(3))/2)))*(x - 7^(1/4)*cos(atan(3*sqrt(3))/2) + i*7^(1/4)*sin(atan(3*sqrt(3))/2)))*(x - 7^(1/4)*cos(atan(3*sqrt(3))/2) - i*7^(1/4)*sin(atan(3*sqrt(3))/2))
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(- y^{4} + y^{2}\right) - 7$$
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
$$a = -1$$
$$b = 1$$
$$c = -7$$
Entonces
$$m = - \frac{1}{2}$$
$$n = - \frac{27}{4}$$
Pues,
$$- \left(y^{2} - \frac{1}{2}\right)^{2} - \frac{27}{4}$$
$$\left(- y^{4} + y^{2}\right) - 7$$
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
$$a = -1$$
$$b = 1$$
$$c = -7$$
Entonces
$$m = - \frac{1}{2}$$
$$n = - \frac{27}{4}$$
Pues,
$$- \left(y^{2} - \frac{1}{2}\right)^{2} - \frac{27}{4}$$
Unión de expresiones racionales
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2 / 2\ -7 + y *\1 - y /
$$y^{2} \left(1 - y^{2}\right) - 7$$
-7 + y^2*(1 - y^2)