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