Sr Examen
Descomponer y^4-5*y^2+10 al cuadrado
Solución
Factorización
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$$\left(x + \left(\sqrt[4]{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{2} \right)} - \sqrt[4]{10} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{2} \right)} + \sqrt[4]{10} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{2} \right)} + \sqrt[4]{10} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{2} \right)} - \sqrt[4]{10} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{2} \right)}\right)\right)$$
(((x + 10^(1/4)*cos(atan(sqrt(15)/5)/2) + i*10^(1/4)*sin(atan(sqrt(15)/5)/2))*(x + 10^(1/4)*cos(atan(sqrt(15)/5)/2) - i*10^(1/4)*sin(atan(sqrt(15)/5)/2)))*(x - 10^(1/4)*cos(atan(sqrt(15)/5)/2) + i*10^(1/4)*sin(atan(sqrt(15)/5)/2)))*(x - 10^(1/4)*cos(atan(sqrt(15)/5)/2) - i*10^(1/4)*sin(atan(sqrt(15)/5)/2))
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(y^{4} - 5 y^{2}\right) + 10$$
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
$$a = 1$$
$$b = -5$$
$$c = 10$$
Entonces
$$m = - \frac{5}{2}$$
$$n = \frac{15}{4}$$
Pues,
$$\left(y^{2} - \frac{5}{2}\right)^{2} + \frac{15}{4}$$
$$\left(y^{4} - 5 y^{2}\right) + 10$$
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
$$a = 1$$
$$b = -5$$
$$c = 10$$
Entonces
$$m = - \frac{5}{2}$$
$$n = \frac{15}{4}$$
Pues,
$$\left(y^{2} - \frac{5}{2}\right)^{2} + \frac{15}{4}$$
Unión de expresiones racionales
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2 / 2\ 10 + y *\-5 + y /
$$y^{2} \left(y^{2} - 5\right) + 10$$
10 + y^2*(-5 + y^2)