Sr Examen
Descomponer 5*x^4+3*x^2+2 al cuadrado
Solución
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(5 x^{4} + 3 x^{2}\right) + 2$$
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
$$a = 5$$
$$b = 3$$
$$c = 2$$
Entonces
$$m = \frac{3}{10}$$
$$n = \frac{31}{20}$$
Pues,
$$5 \left(x^{2} + \frac{3}{10}\right)^{2} + \frac{31}{20}$$
$$\left(5 x^{4} + 3 x^{2}\right) + 2$$
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
$$a = 5$$
$$b = 3$$
$$c = 2$$
Entonces
$$m = \frac{3}{10}$$
$$n = \frac{31}{20}$$
Pues,
$$5 \left(x^{2} + \frac{3}{10}\right)^{2} + \frac{31}{20}$$
Factorización
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$$\left(x + \left(\frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(x + \left(\frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(x + \left(- \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(x + \left(- \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5}\right)\right)$$
(((x + 2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5 + i*2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5)*(x + 2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5 - i*2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5))*(x - 2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5 + i*2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5))*(x - 2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5 - i*2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5)
Unión de expresiones racionales
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2 / 2\ 2 + x *\3 + 5*x /
$$x^{2} \left(5 x^{2} + 3\right) + 2$$
2 + x^2*(3 + 5*x^2)