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