Sr Examen
Factorizar el polinomio a^7+b^7
Solución
Factorización
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$$\left(a - b \left(\cos{\left(\frac{\pi}{7} \right)} + i \sin{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(- \cos^{2}{\left(\frac{\pi}{7} \right)} - \sin^{2}{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(- \cos^{2}{\left(\frac{\pi}{7} \right)} + \sin^{2}{\left(\frac{\pi}{7} \right)} - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(- \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(\sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} - i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right)\right) \left(a - b \left(- \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} - i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(- \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right)\right)$$
((((((a - b*(i*sin(pi/7) + cos(pi/7)))*(a - b*(-cos(pi/7)^2 - sin(pi/7)^2)))*(a - b*(sin(pi/7)^2 - cos(pi/7)^2 - 2*i*cos(pi/7)*sin(pi/7))))*(a - b*(cos(pi/7)*cos(2*pi/7) - sin(pi/7)*sin(2*pi/7) + i*cos(pi/7)*sin(2*pi/7) + i*cos(2*pi/7)*sin(pi/7))))*(a - b*(cos(pi/7)*cos(2*pi/7) + sin(pi/7)*sin(2*pi/7) + i*cos(2*pi/7)*sin(pi/7) - i*cos(pi/7)*sin(2*pi/7))))*(a - b*(-cos(pi/7)*cos(3*pi/7) - sin(pi/7)*sin(3*pi/7) + i*cos(pi/7)*sin(3*pi/7) - i*cos(3*pi/7)*sin(pi/7))))*(a - b*(sin(pi/7)*sin(3*pi/7) - cos(pi/7)*cos(3*pi/7) - i*cos(pi/7)*sin(3*pi/7) - i*cos(3*pi/7)*sin(pi/7)))
Combinatoria
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/ 6 6 2 4 4 2 5 5 3 3\ (a + b)*\a + b + a *b + a *b - a*b - b*a - a *b /
$$\left(a + b\right) \left(a^{6} - a^{5} b + a^{4} b^{2} - a^{3} b^{3} + a^{2} b^{4} - a b^{5} + b^{6}\right)$$
(a + b)*(a^6 + b^6 + a^2*b^4 + a^4*b^2 - a*b^5 - b*a^5 - a^3*b^3)