Sr Examen
Descomponer y^4-y^2+15 al cuadrado
Solución
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(y^{4} - y^{2}\right) + 15$$
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
$$a = 1$$
$$b = -1$$
$$c = 15$$
Entonces
$$m = - \frac{1}{2}$$
$$n = \frac{59}{4}$$
Pues,
$$\left(y^{2} - \frac{1}{2}\right)^{2} + \frac{59}{4}$$
$$\left(y^{4} - y^{2}\right) + 15$$
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
$$a = 1$$
$$b = -1$$
$$c = 15$$
Entonces
$$m = - \frac{1}{2}$$
$$n = \frac{59}{4}$$
Pues,
$$\left(y^{2} - \frac{1}{2}\right)^{2} + \frac{59}{4}$$
Factorización
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/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ | 4 ____ |atan\\/ 59 /| 4 ____ |atan\\/ 59 /|| | 4 ____ |atan\\/ 59 /| 4 ____ |atan\\/ 59 /|| | 4 ____ |atan\\/ 59 /| 4 ____ |atan\\/ 59 /|| | 4 ____ |atan\\/ 59 /| 4 ____ |atan\\/ 59 /|| |x + \/ 15 *cos|------------| + I*\/ 15 *sin|------------||*|x + \/ 15 *cos|------------| - I*\/ 15 *sin|------------||*|x + - \/ 15 *cos|------------| + I*\/ 15 *sin|------------||*|x + - \/ 15 *cos|------------| - I*\/ 15 *sin|------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(x + \left(\sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{59} \right)}}{2} \right)} - \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{59} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{59} \right)}}{2} \right)} + \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{59} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{59} \right)}}{2} \right)} + \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{59} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{59} \right)}}{2} \right)} - \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{59} \right)}}{2} \right)}\right)\right)$$
(((x + 15^(1/4)*cos(atan(sqrt(59))/2) + i*15^(1/4)*sin(atan(sqrt(59))/2))*(x + 15^(1/4)*cos(atan(sqrt(59))/2) - i*15^(1/4)*sin(atan(sqrt(59))/2)))*(x - 15^(1/4)*cos(atan(sqrt(59))/2) + i*15^(1/4)*sin(atan(sqrt(59))/2)))*(x - 15^(1/4)*cos(atan(sqrt(59))/2) - i*15^(1/4)*sin(atan(sqrt(59))/2))
Unión de expresiones racionales
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2 / 2\ 15 + y *\-1 + y /
$$y^{2} \left(y^{2} - 1\right) + 15$$
15 + y^2*(-1 + y^2)