Sr Examen
Descomponer -6*p^4+6*p^2-2 al cuadrado
Solución
Ha introducido
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4 2 - 6*p + 6*p - 2
$$\left(- 6 p^{4} + 6 p^{2}\right) - 2$$
-6*p^4 + 6*p^2 - 2
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(- 6 p^{4} + 6 p^{2}\right) - 2$$
Para eso usemos la fórmula
$$a p^{4} + b p^{2} + c = a \left(m + p^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = -6$$
$$b = 6$$
$$c = -2$$
Entonces
$$m = - \frac{1}{2}$$
$$n = - \frac{1}{2}$$
Pues,
$$- 6 \left(p^{2} - \frac{1}{2}\right)^{2} - \frac{1}{2}$$
$$\left(- 6 p^{4} + 6 p^{2}\right) - 2$$
Para eso usemos la fórmula
$$a p^{4} + b p^{2} + c = a \left(m + p^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = -6$$
$$b = 6$$
$$c = -2$$
Entonces
$$m = - \frac{1}{2}$$
$$n = - \frac{1}{2}$$
Pues,
$$- 6 \left(p^{2} - \frac{1}{2}\right)^{2} - \frac{1}{2}$$
Factorización
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/ ___ 4 ___ ___ 3/4 ___ 4 ___ ___ 3/4\ / ___ 4 ___ ___ 3/4 ___ 3/4 ___ 4 ___\ / ___ 4 ___ ___ 3/4 ___ 4 ___ ___ 3/4\ / ___ 4 ___ ___ 3/4 ___ 3/4 ___ 4 ___\ | \/ 2 *\/ 3 \/ 2 *3 I*\/ 2 *\/ 3 I*\/ 2 *3 | | \/ 2 *\/ 3 \/ 2 *3 I*\/ 2 *3 I*\/ 2 *\/ 3 | | \/ 2 *\/ 3 \/ 2 *3 I*\/ 2 *\/ 3 I*\/ 2 *3 | | \/ 2 *\/ 3 \/ 2 *3 I*\/ 2 *3 I*\/ 2 *\/ 3 | |p + ----------- + ---------- + ------------- - ------------|*|p + ----------- + ---------- + ------------ - -------------|*|p + - ----------- - ---------- + ------------- - ------------|*|p + - ----------- - ---------- + ------------ - -------------| \ 4 12 4 12 / \ 4 12 12 4 / \ 4 12 4 12 / \ 4 12 12 4 /
$$\left(p + \left(\frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{12} + \frac{\sqrt{2} \sqrt[4]{3}}{4} - \frac{\sqrt{2} \sqrt[4]{3} i}{4} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{12}\right)\right) \left(p + \left(\frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{12} + \frac{\sqrt{2} \sqrt[4]{3}}{4} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{12} + \frac{\sqrt{2} \sqrt[4]{3} i}{4}\right)\right) \left(p + \left(- \frac{\sqrt{2} \sqrt[4]{3}}{4} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{12} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{12} + \frac{\sqrt{2} \sqrt[4]{3} i}{4}\right)\right) \left(p + \left(- \frac{\sqrt{2} \sqrt[4]{3}}{4} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}}}{12} - \frac{\sqrt{2} \sqrt[4]{3} i}{4} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} i}{12}\right)\right)$$
(((p + sqrt(2)*3^(1/4)/4 + sqrt(2)*3^(3/4)/12 + i*sqrt(2)*3^(1/4)/4 - i*sqrt(2)*3^(3/4)/12)*(p + sqrt(2)*3^(1/4)/4 + sqrt(2)*3^(3/4)/12 + i*sqrt(2)*3^(3/4)/12 - i*sqrt(2)*3^(1/4)/4))*(p - sqrt(2)*3^(1/4)/4 - sqrt(2)*3^(3/4)/12 + i*sqrt(2)*3^(1/4)/4 - i*sqrt(2)*3^(3/4)/12))*(p - sqrt(2)*3^(1/4)/4 - sqrt(2)*3^(3/4)/12 + i*sqrt(2)*3^(3/4)/12 - i*sqrt(2)*3^(1/4)/4)
Unión de expresiones racionales
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/ 2 / 2\\ 2*\-1 + 3*p *\1 - p //
$$2 \left(3 p^{2} \left(1 - p^{2}\right) - 1\right)$$
2*(-1 + 3*p^2*(1 - p^2))