Mister Exam

Factor polynomial a^7+b^7

An expression to simplify:

The solution

You have entered [src]
 7    7
a  + b 
$$a^{7} + b^{7}$$
a^7 + b^7
Factorization [src]
/      /     /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\\\
|a - b*|I*sin|--| + cos|--|||*|a - b*|- cos |--| - sin |--|||*|a - b*|sin |--| - cos |--| - 2*I*cos|--|*sin|--|||*|a - b*|cos|--|*cos|----| - sin|--|*sin|----| + I*cos|--|*sin|----| + I*cos|----|*sin|--|||*|a - b*|cos|--|*cos|----| + sin|--|*sin|----| + I*cos|----|*sin|--| - I*cos|--|*sin|----|||*|a - b*|- cos|--|*cos|----| - sin|--|*sin|----| + I*cos|--|*sin|----| - I*cos|----|*sin|--|||*|a - b*|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 ///
$$\left(a - b \left(\cos{\left(\frac{\pi}{7} \right)} + i \sin{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(- \cos^{2}{\left(\frac{\pi}{7} \right)} - \sin^{2}{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(- \cos^{2}{\left(\frac{\pi}{7} \right)} + \sin^{2}{\left(\frac{\pi}{7} \right)} - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(- \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(\sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} - i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right)\right) \left(a - b \left(- \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} - i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(a - b \left(- \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right)\right)$$
((((((a - b*(i*sin(pi/7) + cos(pi/7)))*(a - b*(-cos(pi/7)^2 - sin(pi/7)^2)))*(a - b*(sin(pi/7)^2 - cos(pi/7)^2 - 2*i*cos(pi/7)*sin(pi/7))))*(a - b*(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))))*(a - b*(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))))*(a - b*(-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))))*(a - b*(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)))
Numerical answer [src]
a^7 + b^7
a^7 + b^7
Combinatorics [src]
        / 6    6    2  4    4  2      5      5    3  3\
(a + b)*\a  + b  + a *b  + a *b  - a*b  - b*a  - a *b /
$$\left(a + b\right) \left(a^{6} - a^{5} b + a^{4} b^{2} - a^{3} b^{3} + a^{2} b^{4} - a b^{5} + b^{6}\right)$$
(a + b)*(a^6 + b^6 + a^2*b^4 + a^4*b^2 - a*b^5 - b*a^5 - a^3*b^3)
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