Sr Examen
Descomponer x^4-x^2+5 al cuadrado
Solución
Factorización
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$$\left(x + \left(\sqrt[4]{5} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} - \sqrt[4]{5} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{5} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} + \sqrt[4]{5} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{5} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} + \sqrt[4]{5} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{5} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} - \sqrt[4]{5} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right)$$
(((x + 5^(1/4)*cos(atan(sqrt(19))/2) + i*5^(1/4)*sin(atan(sqrt(19))/2))*(x + 5^(1/4)*cos(atan(sqrt(19))/2) - i*5^(1/4)*sin(atan(sqrt(19))/2)))*(x - 5^(1/4)*cos(atan(sqrt(19))/2) + i*5^(1/4)*sin(atan(sqrt(19))/2)))*(x - 5^(1/4)*cos(atan(sqrt(19))/2) - i*5^(1/4)*sin(atan(sqrt(19))/2))
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(x^{4} - x^{2}\right) + 5$$
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
$$a = 1$$
$$b = -1$$
$$c = 5$$
Entonces
$$m = - \frac{1}{2}$$
$$n = \frac{19}{4}$$
Pues,
$$\left(x^{2} - \frac{1}{2}\right)^{2} + \frac{19}{4}$$
$$\left(x^{4} - x^{2}\right) + 5$$
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
$$a = 1$$
$$b = -1$$
$$c = 5$$
Entonces
$$m = - \frac{1}{2}$$
$$n = \frac{19}{4}$$
Pues,
$$\left(x^{2} - \frac{1}{2}\right)^{2} + \frac{19}{4}$$
Unión de expresiones racionales
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2 / 2\ 5 + x *\-1 + x /
$$x^{2} \left(x^{2} - 1\right) + 5$$
5 + x^2*(-1 + x^2)