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