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