Sr Examen
Factorizar el polinomio x^9-y^9
Solución
Factorización
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/ / ___\\ / / ___\\
| | 1 I*\/ 3 || | | 1 I*\/ 3 || / / /pi\ /pi\\\ / / /pi\ /pi\\\ / / /2*pi\ /2*pi\\\ / / /2*pi\ /2*pi\\\ / / /4*pi\ /4*pi\\\ / / /4*pi\ /4*pi\\\
(x - y)*|x - y*|- - - -------||*|x - y*|- - + -------||*|x - y*|- cos|--| - I*sin|--|||*|x - y*|- cos|--| + I*sin|--|||*|x - y*|- I*sin|----| + cos|----|||*|x - y*|I*sin|----| + cos|----|||*|x - y*|- I*sin|----| + cos|----|||*|x - y*|I*sin|----| + cos|----|||
\ \ 2 2 // \ \ 2 2 // \ \ \9 / \9 /// \ \ \9 / \9 /// \ \ \ 9 / \ 9 /// \ \ \ 9 / \ 9 /// \ \ \ 9 / \ 9 /// \ \ \ 9 / \ 9 ///
$$\left(x - y\right) \left(x - y \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)\right) \left(x - y \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)\right) \left(x - y \left(- \cos{\left(\frac{\pi}{9} \right)} - i \sin{\left(\frac{\pi}{9} \right)}\right)\right) \left(x - y \left(- \cos{\left(\frac{\pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)}\right)\right) \left(x - y \left(\cos{\left(\frac{2 \pi}{9} \right)} - i \sin{\left(\frac{2 \pi}{9} \right)}\right)\right) \left(x - y \left(\cos{\left(\frac{2 \pi}{9} \right)} + i \sin{\left(\frac{2 \pi}{9} \right)}\right)\right) \left(x - y \left(\cos{\left(\frac{4 \pi}{9} \right)} - i \sin{\left(\frac{4 \pi}{9} \right)}\right)\right) \left(x - y \left(\cos{\left(\frac{4 \pi}{9} \right)} + i \sin{\left(\frac{4 \pi}{9} \right)}\right)\right)$$
((((((((x - y)*(x - y*(-1/2 - i*sqrt(3)/2)))*(x - y*(-1/2 + i*sqrt(3)/2)))*(x - y*(-cos(pi/9) - i*sin(pi/9))))*(x - y*(-cos(pi/9) + i*sin(pi/9))))*(x - y*(-i*sin(2*pi/9) + cos(2*pi/9))))*(x - y*(i*sin(2*pi/9) + cos(2*pi/9))))*(x - y*(-i*sin(4*pi/9) + cos(4*pi/9))))*(x - y*(i*sin(4*pi/9) + cos(4*pi/9)))
Combinatoria
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/ 2 2 \ / 6 6 3 3\ (x - y)*\x + y + x*y/*\x + y + x *y /
$$\left(x - y\right) \left(x^{2} + x y + y^{2}\right) \left(x^{6} + x^{3} y^{3} + y^{6}\right)$$
(x - y)*(x^2 + y^2 + x*y)*(x^6 + y^6 + x^3*y^3)