Sr Examen
Descomponer -x^4+2*x^2-3 al cuadrado
Solución
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(- x^{4} + 2 x^{2}\right) - 3$$
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
$$a = -1$$
$$b = 2$$
$$c = -3$$
Entonces
$$m = -1$$
$$n = -2$$
Pues,
$$- \left(x^{2} - 1\right)^{2} - 2$$
$$\left(- x^{4} + 2 x^{2}\right) - 3$$
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
$$a = -1$$
$$b = 2$$
$$c = -3$$
Entonces
$$m = -1$$
$$n = -2$$
Pues,
$$- \left(x^{2} - 1\right)^{2} - 2$$
Factorización
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/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | 4 ___ |atan\\/ 2 /| 4 ___ |atan\\/ 2 /|| | 4 ___ |atan\\/ 2 /| 4 ___ |atan\\/ 2 /|| | 4 ___ |atan\\/ 2 /| 4 ___ |atan\\/ 2 /|| | 4 ___ |atan\\/ 2 /| 4 ___ |atan\\/ 2 /|| |x + \/ 3 *cos|-----------| + I*\/ 3 *sin|-----------||*|x + \/ 3 *cos|-----------| - I*\/ 3 *sin|-----------||*|x + - \/ 3 *cos|-----------| + I*\/ 3 *sin|-----------||*|x + - \/ 3 *cos|-----------| - I*\/ 3 *sin|-----------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(x + \left(\sqrt[4]{3} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} - \sqrt[4]{3} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{3} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} + \sqrt[4]{3} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} + \sqrt[4]{3} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} - \sqrt[4]{3} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right)$$
(((x + 3^(1/4)*cos(atan(sqrt(2))/2) + i*3^(1/4)*sin(atan(sqrt(2))/2))*(x + 3^(1/4)*cos(atan(sqrt(2))/2) - i*3^(1/4)*sin(atan(sqrt(2))/2)))*(x - 3^(1/4)*cos(atan(sqrt(2))/2) + i*3^(1/4)*sin(atan(sqrt(2))/2)))*(x - 3^(1/4)*cos(atan(sqrt(2))/2) - i*3^(1/4)*sin(atan(sqrt(2))/2))
Unión de expresiones racionales
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2 / 2\ -3 + x *\2 - x /
$$x^{2} \left(2 - x^{2}\right) - 3$$
-3 + x^2*(2 - x^2)