Sr Examen
Descomponer x^4+2*x^2+8 al cuadrado
Solución
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(x^{4} + 2 x^{2}\right) + 8$$
Para eso usemos la fórmula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = 1$$
$$b = 2$$
$$c = 8$$
Entonces
$$m = 1$$
$$n = 7$$
Pues,
$$\left(x^{2} + 1\right)^{2} + 7$$
$$\left(x^{4} + 2 x^{2}\right) + 8$$
Para eso usemos la fórmula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = 1$$
$$b = 2$$
$$c = 8$$
Entonces
$$m = 1$$
$$n = 7$$
Pues,
$$\left(x^{2} + 1\right)^{2} + 7$$
Factorización
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/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | 3/4 |atan\\/ 7 /| 3/4 |atan\\/ 7 /|| | 3/4 |atan\\/ 7 /| 3/4 |atan\\/ 7 /|| | 3/4 |atan\\/ 7 /| 3/4 |atan\\/ 7 /|| | 3/4 |atan\\/ 7 /| 3/4 |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 //
$$\left(x + \left(2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}\right)\right)$$
(((x + 2^(3/4)*sin(atan(sqrt(7))/2) + i*2^(3/4)*cos(atan(sqrt(7))/2))*(x + 2^(3/4)*sin(atan(sqrt(7))/2) - i*2^(3/4)*cos(atan(sqrt(7))/2)))*(x - 2^(3/4)*sin(atan(sqrt(7))/2) + i*2^(3/4)*cos(atan(sqrt(7))/2)))*(x - 2^(3/4)*sin(atan(sqrt(7))/2) - i*2^(3/4)*cos(atan(sqrt(7))/2))
Unión de expresiones racionales
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2 / 2\ 8 + x *\2 + x /
$$x^{2} \left(x^{2} + 2\right) + 8$$
8 + x^2*(2 + x^2)