Господин Экзамен
Разложить многочлен на множители 128*a^7+b^7
Решение
Разложение на множители
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/ / /pi\ /pi\\\ / / 2/pi\ 2/pi\\\ / / 2/pi\ 2/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\\\ | |cos|--| I*sin|--||| | | cos |--| sin |--||| | |sin |--| cos |--| || | |cos|--|*cos|----| sin|--|*sin|----| I*cos|--|*sin|----| I*cos|----|*sin|--||| | |cos|--|*cos|----| sin|--|*sin|----| I*cos|----|*sin|--| I*cos|--|*sin|----||| | | cos|--|*cos|----| sin|--|*sin|----| I*cos|--|*sin|----| I*cos|----|*sin|--||| | |sin|--|*sin|----| cos|--|*cos|----| I*cos|--|*sin|----| I*cos|----|*sin|--||| | | \7 / \7 /|| | | \7 / \7 /|| | | \7 / \7 / /pi\ /pi\|| | | \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 /|| 1*|a - b*|------- + ---------||*|a - b*|- -------- - --------||*|a - b*|-------- - -------- - I*cos|--|*sin|--|||*|a - b*|----------------- - ----------------- + ------------------- + -------------------||*|a - b*|----------------- + ----------------- + ------------------- - -------------------||*|a - b*|- ----------------- - ----------------- + ------------------- - -------------------||*|a - b*|----------------- - ----------------- - ------------------- - -------------------|| \ \ 2 2 // \ \ 2 2 // \ \ 2 2 \7 / \7 /// \ \ 2 2 2 2 // \ \ 2 2 2 2 // \ \ 2 2 2 2 // \ \ 2 2 2 2 //
$$1 \left(a - b \left(\frac{\cos{\left(\frac{\pi}{7} \right)}}{2} + \frac{i \sin{\left(\frac{\pi}{7} \right)}}{2}\right)\right) \left(a - b \left(- \frac{\cos^{2}{\left(\frac{\pi}{7} \right)}}{2} - \frac{\sin^{2}{\left(\frac{\pi}{7} \right)}}{2}\right)\right) \left(a - b \left(- \frac{\cos^{2}{\left(\frac{\pi}{7} \right)}}{2} + \frac{\sin^{2}{\left(\frac{\pi}{7} \right)}}{2} - i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(- \frac{\sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}}{2} + \frac{i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}}{2} + \frac{i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}}{2}\right)\right) \left(a - b \left(\frac{\sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}}{2} - \frac{i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}}{2} + \frac{i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}}{2}\right)\right) \left(a - b \left(- \frac{\sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)}}{2} - \frac{\cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}}{2} - \frac{i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}}{2} + \frac{i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}}{2}\right)\right) \left(a - b \left(- \frac{\cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)}}{2} - \frac{i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}}{2} - \frac{i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}}{2}\right)\right)$$
((((((1*(a - b*(cos(pi/7)/2 + i*sin(pi/7)/2)))*(a - b*(-cos(pi/7)^2/2 - sin(pi/7)^2/2)))*(a - b*(sin(pi/7)^2/2 - cos(pi/7)^2/2 - i*cos(pi/7)*sin(pi/7))))*(a - b*(cos(pi/7)*cos(2*pi/7)/2 - sin(pi/7)*sin(2*pi/7)/2 + i*cos(pi/7)*sin(2*pi/7)/2 + i*cos(2*pi/7)*sin(pi/7)/2)))*(a - b*(cos(pi/7)*cos(2*pi/7)/2 + sin(pi/7)*sin(2*pi/7)/2 + i*cos(2*pi/7)*sin(pi/7)/2 - i*cos(pi/7)*sin(2*pi/7)/2)))*(a - b*(-cos(pi/7)*cos(3*pi/7)/2 - sin(pi/7)*sin(3*pi/7)/2 + i*cos(pi/7)*sin(3*pi/7)/2 - i*cos(3*pi/7)*sin(pi/7)/2)))*(a - b*(sin(pi/7)*sin(3*pi/7)/2 - cos(pi/7)*cos(3*pi/7)/2 - i*cos(pi/7)*sin(3*pi/7)/2 - i*cos(3*pi/7)*sin(pi/7)/2))
Комбинаторика
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/ 6 6 5 3 3 5 2 4 4 2\ (b + 2*a)*\b + 64*a - 32*b*a - 8*a *b - 2*a*b + 4*a *b + 16*a *b /
$$\left(2 a + b\right) \left(64 a^{6} - 32 a^{5} b + 16 a^{4} b^{2} - 8 a^{3} b^{3} + 4 a^{2} b^{4} - 2 a b^{5} + b^{6}\right)$$
(b + 2*a)*(b^6 + 64*a^6 - 32*b*a^5 - 8*a^3*b^3 - 2*a*b^5 + 4*a^2*b^4 + 16*a^4*b^2)