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