Господин Экзамен
Разложить многочлен на множители x^7-1
Решение
Разложение на множители
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/ /pi\ /pi\\ / /pi\ /pi\\ / /2*pi\ /2*pi\\ / /2*pi\ /2*pi\\ / /3*pi\ /3*pi\\ / /3*pi\ /3*pi\\
1*(x - 1)*|x + cos|--| + I*sin|--||*|x + cos|--| - I*sin|--||*|x + I*sin|----| - cos|----||*|x + - I*sin|----| - cos|----||*|x + cos|----| + I*sin|----||*|x + cos|----| - I*sin|----||
\ \7 / \7 // \ \7 / \7 // \ \ 7 / \ 7 // \ \ 7 / \ 7 // \ \ 7 / \ 7 // \ \ 7 / \ 7 //
$$1 \left(x - 1\right) \left(x + \left(\cos{\left(\frac{\pi}{7} \right)} + i \sin{\left(\frac{\pi}{7} \right)}\right)\right) \left(x + \left(\cos{\left(\frac{\pi}{7} \right)} - i \sin{\left(\frac{\pi}{7} \right)}\right)\right) \left(x - \left(\cos{\left(\frac{2 \pi}{7} \right)} - i \sin{\left(\frac{2 \pi}{7} \right)}\right)\right) \left(x - \left(\cos{\left(\frac{2 \pi}{7} \right)} + i \sin{\left(\frac{2 \pi}{7} \right)}\right)\right) \left(x + \left(\cos{\left(\frac{3 \pi}{7} \right)} + i \sin{\left(\frac{3 \pi}{7} \right)}\right)\right) \left(x + \left(\cos{\left(\frac{3 \pi}{7} \right)} - i \sin{\left(\frac{3 \pi}{7} \right)}\right)\right)$$
((((((1*(x - 1))*(x + (cos(pi/7) + i*sin(pi/7))))*(x + (cos(pi/7) - i*sin(pi/7))))*(x + (i*sin(2*pi/7) - cos(2*pi/7))))*(x - (i*sin(2*pi/7) - cos(2*pi/7))))*(x + (cos(3*pi/7) + i*sin(3*pi/7))))*(x + (cos(3*pi/7) - i*sin(3*pi/7)))
Комбинаторика
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/ 2 3 4 5 6\ (-1 + x)*\1 + x + x + x + x + x + x /
$$\left(x - 1\right) \left(x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1\right)$$
(-1 + x)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)