Sr Examen
Descomponer y^4+4*y^2+10 al cuadrado
Solución
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(y^{4} + 4 y^{2}\right) + 10$$
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
$$a = 1$$
$$b = 4$$
$$c = 10$$
Entonces
$$m = 2$$
$$n = 6$$
Pues,
$$\left(y^{2} + 2\right)^{2} + 6$$
$$\left(y^{4} + 4 y^{2}\right) + 10$$
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
$$a = 1$$
$$b = 4$$
$$c = 10$$
Entonces
$$m = 2$$
$$n = 6$$
Pues,
$$\left(y^{2} + 2\right)^{2} + 6$$
Factorización
[src]
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | | |\/ 6 || | |\/ 6 ||| | | |\/ 6 || | |\/ 6 ||| | | |\/ 6 || | |\/ 6 ||| | | |\/ 6 || | |\/ 6 ||| | |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----||| | 4 ____ | \ 2 /| 4 ____ | \ 2 /|| | 4 ____ | \ 2 /| 4 ____ | \ 2 /|| | 4 ____ | \ 2 /| 4 ____ | \ 2 /|| | 4 ____ | \ 2 /| 4 ____ | \ 2 /|| |x + \/ 10 *sin|-----------| + I*\/ 10 *cos|-----------||*|x + \/ 10 *sin|-----------| - I*\/ 10 *cos|-----------||*|x + - \/ 10 *sin|-----------| + I*\/ 10 *cos|-----------||*|x + - \/ 10 *sin|-----------| - I*\/ 10 *cos|-----------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(x + \left(\sqrt[4]{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{2} \right)} - \sqrt[4]{10} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{2} \right)} + \sqrt[4]{10} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{2} \right)} + \sqrt[4]{10} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{2} \right)} - \sqrt[4]{10} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{2} \right)}\right)\right)$$
(((x + 10^(1/4)*sin(atan(sqrt(6)/2)/2) + i*10^(1/4)*cos(atan(sqrt(6)/2)/2))*(x + 10^(1/4)*sin(atan(sqrt(6)/2)/2) - i*10^(1/4)*cos(atan(sqrt(6)/2)/2)))*(x - 10^(1/4)*sin(atan(sqrt(6)/2)/2) + i*10^(1/4)*cos(atan(sqrt(6)/2)/2)))*(x - 10^(1/4)*sin(atan(sqrt(6)/2)/2) - i*10^(1/4)*cos(atan(sqrt(6)/2)/2))
Unión de expresiones racionales
[src]
2 / 2\ 10 + y *\4 + y /
$$y^{2} \left(y^{2} + 4\right) + 10$$
10 + y^2*(4 + y^2)