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Expression simplification/ Factorization polynomials/ x^39+1

Factor polynomial x^39+1

An expression to simplify:

The solution

You have entered [src] 
 39    
x   + 1
x39+1x^{39} + 1 
x^39 + 1
Factorization [src] 
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/           /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\    /4*pi\      /pi\    /4*pi\        /pi\    /4*pi\        /4*pi\    /pi\\ /         /pi\    /4*pi\      /pi\    /4*pi\        /4*pi\    /pi\        /pi\    /4*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\    /8*pi\      /pi\    /8*pi\        /pi\    /8*pi\        /8*pi\    /pi\\ /         /pi\    /8*pi\      /pi\    /8*pi\        /8*pi\    /pi\        /pi\    /8*pi\\ /         /pi\    /10*pi\      /pi\    /10*pi\        /pi\    /10*pi\        /10*pi\    /pi\\ /         /pi\    /10*pi\      /pi\    /10*pi\        /10*pi\    /pi\        /pi\    /10*pi\\ /         /pi\    /4*pi\      /pi\    /4*pi\        /pi\    /4*pi\        /4*pi\    /pi\\ /         /pi\    /4*pi\      /pi\    /4*pi\        /4*pi\    /pi\        /pi\    /4*pi\\ /         /pi\    /14*pi\      /pi\    /14*pi\        /pi\    /14*pi\        /14*pi\    /pi\\ /         /pi\    /14*pi\      /pi\    /14*pi\        /14*pi\    /pi\        /pi\    /14*pi\\ /         /pi\    /16*pi\      /pi\    /16*pi\        /pi\    /16*pi\        /16*pi\    /pi\\ /         /pi\    /16*pi\      /pi\    /16*pi\        /16*pi\    /pi\        /pi\    /16*pi\\ /         /pi\    /6*pi\      /pi\    /6*pi\        /pi\    /6*pi\        /6*pi\    /pi\\ /         /pi\    /6*pi\      /pi\    /6*pi\        /6*pi\    /pi\        /pi\    /6*pi\\ /       /pi\    /19*pi\      /pi\    /19*pi\        /pi\    /19*pi\        /19*pi\    /pi\\ /       /pi\    /pi\      /pi\    /pi\        /pi\    /pi\        /pi\    /pi\\ /         /pi\    /pi\      /pi\    /pi\        /pi\    /pi\        /pi\    /pi\\ /       /pi\    /5*pi\      /pi\    /5*pi\        /pi\    /5*pi\        /5*pi\    /pi\\ /         /pi\    /5*pi\      /pi\    /5*pi\        /pi\    /5*pi\        /5*pi\    /pi\\ /       /pi\    /7*pi\      /pi\    /7*pi\        /pi\    /7*pi\        /7*pi\    /pi\\ /         /pi\    /7*pi\      /pi\    /7*pi\        /pi\    /7*pi\        /7*pi\    /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\\ /       /pi\    /11*pi\      /pi\    /11*pi\        /pi\    /11*pi\        /11*pi\    /pi\\ /         /pi\    /11*pi\      /pi\    /11*pi\        /pi\    /11*pi\        /11*pi\    /pi\\ /       /pi\    /5*pi\      /pi\    /5*pi\        /pi\    /5*pi\        /5*pi\    /pi\\ /         /pi\    /5*pi\      /pi\    /5*pi\        /pi\    /5*pi\        /5*pi\    /pi\\ /       /pi\    /17*pi\      /pi\    /17*pi\        /pi\    /17*pi\        /17*pi\    /pi\\ /         /pi\    /17*pi\      /pi\    /17*pi\        /pi\    /17*pi\        /17*pi\    /pi\\ /         /pi\    /19*pi\      /pi\    /19*pi\        /pi\    /19*pi\        /19*pi\    /pi\\ |       \39/        \39/            \39/              \39/| |       \39/            \39/        \39/              \39/|
|x + - I*sin|--| - cos|--||*|x + cos |--| + sin |--||*|x + - sin |--| + cos |--| + 2*I*cos|--|*sin|--||*|x + - cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| - I*cos|----|*sin|--||*|x + - cos|--|*cos|----| - sin|--|*sin|----| - I*cos|----|*sin|--| + I*cos|--|*sin|----||*|x + - cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| - I*cos|----|*sin|--||*|x + - cos|--|*cos|----| - sin|--|*sin|----| - I*cos|----|*sin|--| + I*cos|--|*sin|----||*|x + - cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| - I*cos|----|*sin|--||*|x + - cos|--|*cos|----| - sin|--|*sin|----| - I*cos|----|*sin|--| + I*cos|--|*sin|----||*|x + - cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| - I*cos|----|*sin|--||*|x + - cos|--|*cos|----| - sin|--|*sin|----| - I*cos|----|*sin|--| + I*cos|--|*sin|----||*|x + - cos|--|*cos|-----| + sin|--|*sin|-----| - I*cos|--|*sin|-----| - I*cos|-----|*sin|--||*|x + - cos|--|*cos|-----| - sin|--|*sin|-----| - I*cos|-----|*sin|--| + I*cos|--|*sin|-----||*|x + - cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| - I*cos|----|*sin|--||*|x + - cos|--|*cos|----| - sin|--|*sin|----| - I*cos|----|*sin|--| + I*cos|--|*sin|----||*|x + - cos|--|*cos|-----| + sin|--|*sin|-----| - I*cos|--|*sin|-----| - I*cos|-----|*sin|--||*|x + - cos|--|*cos|-----| - sin|--|*sin|-----| - I*cos|-----|*sin|--| + I*cos|--|*sin|-----||*|x + - cos|--|*cos|-----| + sin|--|*sin|-----| - I*cos|--|*sin|-----| - I*cos|-----|*sin|--||*|x + - cos|--|*cos|-----| - sin|--|*sin|-----| - I*cos|-----|*sin|--| + I*cos|--|*sin|-----||*|x + - cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| - I*cos|----|*sin|--||*|x + - cos|--|*cos|----| - sin|--|*sin|----| - I*cos|----|*sin|--| + I*cos|--|*sin|----||*|x + cos|--|*cos|-----| + sin|--|*sin|-----| - I*cos|--|*sin|-----| + I*cos|-----|*sin|--||*|x + cos|--|*cos|--| + sin|--|*sin|--| - I*cos|--|*sin|--| + I*cos|--|*sin|--||*|x + - sin|--|*sin|--| + cos|--|*cos|--| + I*cos|--|*sin|--| + I*cos|--|*sin|--||*|x + cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| + I*cos|----|*sin|--||*|x + - sin|--|*sin|----| + cos|--|*cos|----| + I*cos|--|*sin|----| + I*cos|----|*sin|--||*|x + cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| + I*cos|----|*sin|--||*|x + - sin|--|*sin|----| + cos|--|*cos|----| + I*cos|--|*sin|----| + I*cos|----|*sin|--||*|x + cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| + I*cos|----|*sin|--||*|x + - sin|--|*sin|----| + cos|--|*cos|----| + I*cos|--|*sin|----| + I*cos|----|*sin|--||*|x + cos|--|*cos|-----| + sin|--|*sin|-----| - I*cos|--|*sin|-----| + I*cos|-----|*sin|--||*|x + - sin|--|*sin|-----| + cos|--|*cos|-----| + I*cos|--|*sin|-----| + I*cos|-----|*sin|--||*|x + cos|--|*cos|----| + sin|--|*sin|----| - I*cos|--|*sin|----| + I*cos|----|*sin|--||*|x + - sin|--|*sin|----| + cos|--|*cos|----| + I*cos|--|*sin|----| + I*cos|----|*sin|--||*|x + cos|--|*cos|-----| + sin|--|*sin|-----| - I*cos|--|*sin|-----| + I*cos|-----|*sin|--||*|x + - sin|--|*sin|-----| + cos|--|*cos|-----| + I*cos|--|*sin|-----| + I*cos|-----|*sin|--||*|x + - sin|--|*sin|-----| + cos|--|*cos|-----| + I*cos|--|*sin|-----| + I*cos|-----|*sin|--||*|x + ------- + --------- + ------------- - ---------------|*|x + ------- - ------------- + --------- + ---------------|
\           \39/      \39// \        \39/       \39// \          \39/       \39/          \39/    \39// \         \39/    \ 39 /      \39/    \ 39 /        \39/    \ 39 /        \ 39 /    \39// \         \39/    \ 39 /      \39/    \ 39 /        \ 39 /    \39/        \39/    \ 39 // \         \39/    \ 39 /      \39/    \ 39 /        \39/    \ 39 /        \ 39 /    \39// \         \39/    \ 39 /      \39/    \ 39 /        \ 39 /    \39/        \39/    \ 39 // \         \39/    \ 13 /      \39/    \ 13 /        \39/    \ 13 /        \ 13 /    \39// \         \39/    \ 13 /      \39/    \ 13 /        \ 13 /    \39/        \39/    \ 13 // \         \39/    \ 39 /      \39/    \ 39 /        \39/    \ 39 /        \ 39 /    \39// \         \39/    \ 39 /      \39/    \ 39 /        \ 39 /    \39/        \39/    \ 39 // \         \39/    \  39 /      \39/    \  39 /        \39/    \  39 /        \  39 /    \39// \         \39/    \  39 /      \39/    \  39 /        \  39 /    \39/        \39/    \  39 // \         \39/    \ 13 /      \39/    \ 13 /        \39/    \ 13 /        \ 13 /    \39// \         \39/    \ 13 /      \39/    \ 13 /        \ 13 /    \39/        \39/    \ 13 // \         \39/    \  39 /      \39/    \  39 /        \39/    \  39 /        \  39 /    \39// \         \39/    \  39 /      \39/    \  39 /        \  39 /    \39/        \39/    \  39 // \         \39/    \  39 /      \39/    \  39 /        \39/    \  39 /        \  39 /    \39// \         \39/    \  39 /      \39/    \  39 /        \  39 /    \39/        \39/    \  39 // \         \39/    \ 13 /      \39/    \ 13 /        \39/    \ 13 /        \ 13 /    \39// \         \39/    \ 13 /      \39/    \ 13 /        \ 13 /    \39/        \39/    \ 13 // \       \39/    \  39 /      \39/    \  39 /        \39/    \  39 /        \  39 /    \39// \       \13/    \39/      \13/    \39/        \39/    \13/        \13/    \39// \         \13/    \39/      \13/    \39/        \13/    \39/        \39/    \13// \       \39/    \ 39 /      \39/    \ 39 /        \39/    \ 39 /        \ 39 /    \39// \         \39/    \ 39 /      \39/    \ 39 /        \39/    \ 39 /        \ 39 /    \39// \       \39/    \ 39 /      \39/    \ 39 /        \39/    \ 39 /        \ 39 /    \39// \         \39/    \ 39 /      \39/    \ 39 /        \39/    \ 39 /        \ 39 /    \39// \       \39/    \ 13 /      \39/    \ 13 /        \39/    \ 13 /        \ 13 /    \39// \         \39/    \ 13 /      \39/    \ 13 /        \39/    \ 13 /        \ 13 /    \39// \       \39/    \  39 /      \39/    \  39 /        \39/    \  39 /        \  39 /    \39// \         \39/    \  39 /      \39/    \  39 /        \39/    \  39 /        \  39 /    \39// \       \39/    \ 13 /      \39/    \ 13 /        \39/    \ 13 /        \ 13 /    \39// \         \39/    \ 13 /      \39/    \ 13 /        \39/    \ 13 /        \ 13 /    \39// \       \39/    \  39 /      \39/    \  39 /        \39/    \  39 /        \  39 /    \39// \         \39/    \  39 /      \39/    \  39 /        \39/    \  39 /        \  39 /    \39// \         \39/    \  39 /      \39/    \  39 /        \39/    \  39 /        \  39 /    \39// \       2          2             2                2       / \       2            2             2              2       /
(x+(sin⁡2(π39)+cos⁡2(π39)))(x+(−cos⁡(π39)−isin⁡(π39)))(x+(−sin⁡2(π39)+cos⁡2(π39)+2isin⁡(π39)cos⁡(π39)))(x+(−cos⁡(π39)cos⁡(2π39)+sin⁡(π39)sin⁡(2π39)−isin⁡(2π39)cos⁡(π39)−isin⁡(π39)cos⁡(2π39)))(x+(−cos⁡(π39)cos⁡(2π39)−sin⁡(π39)sin⁡(2π39)−isin⁡(π39)cos⁡(2π39)+isin⁡(2π39)cos⁡(π39)))(x+(−cos⁡(π39)cos⁡(4π39)+sin⁡(π39)sin⁡(4π39)−isin⁡(4π39)cos⁡(π39)−isin⁡(π39)cos⁡(4π39)))(x+(−cos⁡(π39)cos⁡(4π39)−sin⁡(π39)sin⁡(4π39)−isin⁡(π39)cos⁡(4π39)+isin⁡(4π39)cos⁡(π39)))(x+(−cos⁡(π39)cos⁡(2π13)+sin⁡(π39)sin⁡(2π13)−isin⁡(2π13)cos⁡(π39)−isin⁡(π39)cos⁡(2π13)))(x+(−cos⁡(π39)cos⁡(2π13)−sin⁡(π39)sin⁡(2π13)−isin⁡(π39)cos⁡(2π13)+isin⁡(2π13)cos⁡(π39)))(x+(−cos⁡(π39)cos⁡(8π39)+sin⁡(π39)sin⁡(8π39)−isin⁡(8π39)cos⁡(π39)−isin⁡(π39)cos⁡(8π39)))(x+(−cos⁡(π39)cos⁡(8π39)−sin⁡(π39)sin⁡(8π39)−isin⁡(π39)cos⁡(8π39)+isin⁡(8π39)cos⁡(π39)))(x+(−cos⁡(π39)cos⁡(10π39)+sin⁡(π39)sin⁡(10π39)−isin⁡(10π39)cos⁡(π39)−isin⁡(π39)cos⁡(10π39)))(x+(−cos⁡(π39)cos⁡(10π39)−sin⁡(π39)sin⁡(10π39)−isin⁡(π39)cos⁡(10π39)+isin⁡(10π39)cos⁡(π39)))(x+(−cos⁡(π39)cos⁡(4π13)+sin⁡(π39)sin⁡(4π13)−isin⁡(4π13)cos⁡(π39)−isin⁡(π39)cos⁡(4π13)))(x+(−cos⁡(π39)cos⁡(4π13)−sin⁡(π39)sin⁡(4π13)−isin⁡(π39)cos⁡(4π13)+isin⁡(4π13)cos⁡(π39)))(x+(−cos⁡(π39)cos⁡(14π39)+sin⁡(π39)sin⁡(14π39)−isin⁡(14π39)cos⁡(π39)−isin⁡(π39)cos⁡(14π39)))(x+(−cos⁡(π39)cos⁡(14π39)−sin⁡(π39)sin⁡(14π39)−isin⁡(π39)cos⁡(14π39)+isin⁡(14π39)cos⁡(π39)))(x+(−cos⁡(π39)cos⁡(16π39)+sin⁡(π39)sin⁡(16π39)−isin⁡(16π39)cos⁡(π39)−isin⁡(π39)cos⁡(16π39)))(x+(−cos⁡(π39)cos⁡(16π39)−sin⁡(π39)sin⁡(16π39)−isin⁡(π39)cos⁡(16π39)+isin⁡(16π39)cos⁡(π39)))(x+(−cos⁡(π39)cos⁡(6π13)+sin⁡(π39)sin⁡(6π13)−isin⁡(6π13)cos⁡(π39)−isin⁡(π39)cos⁡(6π13)))(x+(−cos⁡(π39)cos⁡(6π13)−sin⁡(π39)sin⁡(6π13)−isin⁡(π39)cos⁡(6π13)+isin⁡(6π13)cos⁡(π39)))(x+(cos⁡(π39)cos⁡(19π39)+sin⁡(π39)sin⁡(19π39)−isin⁡(19π39)cos⁡(π39)+isin⁡(π39)cos⁡(19π39)))(x+(sin⁡(π39)sin⁡(π13)+cos⁡(π39)cos⁡(π13)−isin⁡(π13)cos⁡(π39)+isin⁡(π39)cos⁡(π13)))(x+(−sin⁡(π39)sin⁡(π13)+cos⁡(π39)cos⁡(π13)+isin⁡(π39)cos⁡(π13)+isin⁡(π13)cos⁡(π39)))(x+(sin⁡(π39)sin⁡(5π39)+cos⁡(π39)cos⁡(5π39)−isin⁡(5π39)cos⁡(π39)+isin⁡(π39)cos⁡(5π39)))(x+(−sin⁡(π39)sin⁡(5π39)+cos⁡(π39)cos⁡(5π39)+isin⁡(π39)cos⁡(5π39)+isin⁡(5π39)cos⁡(π39)))(x+(sin⁡(π39)sin⁡(7π39)+cos⁡(π39)cos⁡(7π39)−isin⁡(7π39)cos⁡(π39)+isin⁡(π39)cos⁡(7π39)))(x+(−sin⁡(π39)sin⁡(7π39)+cos⁡(π39)cos⁡(7π39)+isin⁡(π39)cos⁡(7π39)+isin⁡(7π39)cos⁡(π39)))(x+(sin⁡(π39)sin⁡(3π13)+cos⁡(π39)cos⁡(3π13)−isin⁡(3π13)cos⁡(π39)+isin⁡(π39)cos⁡(3π13)))(x+(−sin⁡(π39)sin⁡(3π13)+cos⁡(π39)cos⁡(3π13)+isin⁡(π39)cos⁡(3π13)+isin⁡(3π13)cos⁡(π39)))(x+(sin⁡(π39)sin⁡(11π39)+cos⁡(π39)cos⁡(11π39)−isin⁡(11π39)cos⁡(π39)+isin⁡(π39)cos⁡(11π39)))(x+(−sin⁡(π39)sin⁡(11π39)+cos⁡(π39)cos⁡(11π39)+isin⁡(π39)cos⁡(11π39)+isin⁡(11π39)cos⁡(π39)))(x+(sin⁡(π39)sin⁡(5π13)+cos⁡(π39)cos⁡(5π13)−isin⁡(5π13)cos⁡(π39)+isin⁡(π39)cos⁡(5π13)))(x+(−sin⁡(π39)sin⁡(5π13)+cos⁡(π39)cos⁡(5π13)+isin⁡(π39)cos⁡(5π13)+isin⁡(5π13)cos⁡(π39)))(x+(sin⁡(π39)sin⁡(17π39)+cos⁡(π39)cos⁡(17π39)−isin⁡(17π39)cos⁡(π39)+isin⁡(π39)cos⁡(17π39)))(x+(−sin⁡(π39)sin⁡(17π39)+cos⁡(π39)cos⁡(17π39)+isin⁡(π39)cos⁡(17π39)+isin⁡(17π39)cos⁡(π39)))(x+(−sin⁡(π39)sin⁡(19π39)+cos⁡(π39)cos⁡(19π39)+isin⁡(π39)cos⁡(19π39)+isin⁡(19π39)cos⁡(π39)))(x+(3sin⁡(π39)2+cos⁡(π39)2−3icos⁡(π39)2+isin⁡(π39)2))(x+(−3sin⁡(π39)2+cos⁡(π39)2+isin⁡(π39)2+3icos⁡(π39)2))\left(x + \left(\sin^{2}{\left(\frac{\pi}{39} \right)} + \cos^{2}{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \sin^{2}{\left(\frac{\pi}{39} \right)} + \cos^{2}{\left(\frac{\pi}{39} \right)} + 2 i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{2 \pi}{39} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{2 \pi}{39} \right)} - i \sin{\left(\frac{2 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{2 \pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{2 \pi}{39} \right)} - \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{2 \pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{2 \pi}{39} \right)} + i \sin{\left(\frac{2 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{4 \pi}{39} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{4 \pi}{39} \right)} - i \sin{\left(\frac{4 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{4 \pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{4 \pi}{39} \right)} - \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{4 \pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{4 \pi}{39} \right)} + i \sin{\left(\frac{4 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{2 \pi}{13} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{2 \pi}{13} \right)} - i \sin{\left(\frac{2 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{2 \pi}{13} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{2 \pi}{13} \right)} - \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{2 \pi}{13} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{2 \pi}{13} \right)} + i \sin{\left(\frac{2 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{8 \pi}{39} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{8 \pi}{39} \right)} - i \sin{\left(\frac{8 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{8 \pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{8 \pi}{39} \right)} - \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{8 \pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{8 \pi}{39} \right)} + i \sin{\left(\frac{8 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{10 \pi}{39} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{10 \pi}{39} \right)} - i \sin{\left(\frac{10 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{10 \pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{10 \pi}{39} \right)} - \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{10 \pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{10 \pi}{39} \right)} + i \sin{\left(\frac{10 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{4 \pi}{13} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{4 \pi}{13} \right)} - i \sin{\left(\frac{4 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{4 \pi}{13} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{4 \pi}{13} \right)} - \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{4 \pi}{13} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{4 \pi}{13} \right)} + i \sin{\left(\frac{4 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{14 \pi}{39} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{14 \pi}{39} \right)} - i \sin{\left(\frac{14 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{14 \pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{14 \pi}{39} \right)} - \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{14 \pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{14 \pi}{39} \right)} + i \sin{\left(\frac{14 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{16 \pi}{39} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{16 \pi}{39} \right)} - i \sin{\left(\frac{16 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{16 \pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{16 \pi}{39} \right)} - \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{16 \pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{16 \pi}{39} \right)} + i \sin{\left(\frac{16 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{6 \pi}{13} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{6 \pi}{13} \right)} - i \sin{\left(\frac{6 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{6 \pi}{13} \right)}\right)\right) \left(x + \left(- \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{6 \pi}{13} \right)} - \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{6 \pi}{13} \right)} - i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{6 \pi}{13} \right)} + i \sin{\left(\frac{6 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(\cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{19 \pi}{39} \right)} + \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{19 \pi}{39} \right)} - i \sin{\left(\frac{19 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{19 \pi}{39} \right)}\right)\right) \left(x + \left(\sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{\pi}{13} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{\pi}{13} \right)} - i \sin{\left(\frac{\pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{\pi}{13} \right)}\right)\right) \left(x + \left(- \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{\pi}{13} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{\pi}{13} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{\pi}{13} \right)} + i \sin{\left(\frac{\pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(\sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{5 \pi}{39} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{5 \pi}{39} \right)} - i \sin{\left(\frac{5 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{5 \pi}{39} \right)}\right)\right) \left(x + \left(- \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{5 \pi}{39} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{5 \pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{5 \pi}{39} \right)} + i \sin{\left(\frac{5 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(\sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{7 \pi}{39} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{7 \pi}{39} \right)} - i \sin{\left(\frac{7 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{7 \pi}{39} \right)}\right)\right) \left(x + \left(- \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{7 \pi}{39} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{7 \pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{7 \pi}{39} \right)} + i \sin{\left(\frac{7 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(\sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{3 \pi}{13} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{3 \pi}{13} \right)} - i \sin{\left(\frac{3 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{3 \pi}{13} \right)}\right)\right) \left(x + \left(- \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{3 \pi}{13} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{3 \pi}{13} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{3 \pi}{13} \right)} + i \sin{\left(\frac{3 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(\sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{11 \pi}{39} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{11 \pi}{39} \right)} - i \sin{\left(\frac{11 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{11 \pi}{39} \right)}\right)\right) \left(x + \left(- \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{11 \pi}{39} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{11 \pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{11 \pi}{39} \right)} + i \sin{\left(\frac{11 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(\sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{5 \pi}{13} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{5 \pi}{13} \right)} - i \sin{\left(\frac{5 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{5 \pi}{13} \right)}\right)\right) \left(x + \left(- \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{5 \pi}{13} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{5 \pi}{13} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{5 \pi}{13} \right)} + i \sin{\left(\frac{5 \pi}{13} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(\sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{17 \pi}{39} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{17 \pi}{39} \right)} - i \sin{\left(\frac{17 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{17 \pi}{39} \right)}\right)\right) \left(x + \left(- \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{17 \pi}{39} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{17 \pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{17 \pi}{39} \right)} + i \sin{\left(\frac{17 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(- \sin{\left(\frac{\pi}{39} \right)} \sin{\left(\frac{19 \pi}{39} \right)} + \cos{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{19 \pi}{39} \right)} + i \sin{\left(\frac{\pi}{39} \right)} \cos{\left(\frac{19 \pi}{39} \right)} + i \sin{\left(\frac{19 \pi}{39} \right)} \cos{\left(\frac{\pi}{39} \right)}\right)\right) \left(x + \left(\frac{\sqrt{3} \sin{\left(\frac{\pi}{39} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{39} \right)}}{2} - \frac{\sqrt{3} i \cos{\left(\frac{\pi}{39} \right)}}{2} + \frac{i \sin{\left(\frac{\pi}{39} \right)}}{2}\right)\right) \left(x + \left(- \frac{\sqrt{3} \sin{\left(\frac{\pi}{39} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{39} \right)}}{2} + \frac{i \sin{\left(\frac{\pi}{39} \right)}}{2} + \frac{\sqrt{3} i \cos{\left(\frac{\pi}{39} \right)}}{2}\right)\right) 
((((((((((((((((((((((((((((((((((((((x - i*sin(pi/39) - cos(pi/39))*(x + cos(pi/39)^2 + sin(pi/39)^2))*(x - sin(pi/39)^2 + cos(pi/39)^2 + 2*i*cos(pi/39)*sin(pi/39)))*(x - cos(pi/39)*cos(2*pi/39) + sin(pi/39)*sin(2*pi/39) - i*cos(pi/39)*sin(2*pi/39) - i*cos(2*pi/39)*sin(pi/39)))*(x - cos(pi/39)*cos(2*pi/39) - sin(pi/39)*sin(2*pi/39) - i*cos(2*pi/39)*sin(pi/39) + i*cos(pi/39)*sin(2*pi/39)))*(x - cos(pi/39)*cos(4*pi/39) + sin(pi/39)*sin(4*pi/39) - i*cos(pi/39)*sin(4*pi/39) - i*cos(4*pi/39)*sin(pi/39)))*(x - cos(pi/39)*cos(4*pi/39) - sin(pi/39)*sin(4*pi/39) - i*cos(4*pi/39)*sin(pi/39) + i*cos(pi/39)*sin(4*pi/39)))*(x - cos(pi/39)*cos(2*pi/13) + sin(pi/39)*sin(2*pi/13) - i*cos(pi/39)*sin(2*pi/13) - i*cos(2*pi/13)*sin(pi/39)))*(x - cos(pi/39)*cos(2*pi/13) - sin(pi/39)*sin(2*pi/13) - i*cos(2*pi/13)*sin(pi/39) + i*cos(pi/39)*sin(2*pi/13)))*(x - cos(pi/39)*cos(8*pi/39) + sin(pi/39)*sin(8*pi/39) - i*cos(pi/39)*sin(8*pi/39) - i*cos(8*pi/39)*sin(pi/39)))*(x - cos(pi/39)*cos(8*pi/39) - sin(pi/39)*sin(8*pi/39) - i*cos(8*pi/39)*sin(pi/39) + i*cos(pi/39)*sin(8*pi/39)))*(x - cos(pi/39)*cos(10*pi/39) + sin(pi/39)*sin(10*pi/39) - i*cos(pi/39)*sin(10*pi/39) - i*cos(10*pi/39)*sin(pi/39)))*(x - cos(pi/39)*cos(10*pi/39) - sin(pi/39)*sin(10*pi/39) - i*cos(10*pi/39)*sin(pi/39) + i*cos(pi/39)*sin(10*pi/39)))*(x - cos(pi/39)*cos(4*pi/13) + sin(pi/39)*sin(4*pi/13) - i*cos(pi/39)*sin(4*pi/13) - i*cos(4*pi/13)*sin(pi/39)))*(x - cos(pi/39)*cos(4*pi/13) - sin(pi/39)*sin(4*pi/13) - i*cos(4*pi/13)*sin(pi/39) + i*cos(pi/39)*sin(4*pi/13)))*(x - cos(pi/39)*cos(14*pi/39) + sin(pi/39)*sin(14*pi/39) - i*cos(pi/39)*sin(14*pi/39) - i*cos(14*pi/39)*sin(pi/39)))*(x - cos(pi/39)*cos(14*pi/39) - sin(pi/39)*sin(14*pi/39) - i*cos(14*pi/39)*sin(pi/39) + i*cos(pi/39)*sin(14*pi/39)))*(x - cos(pi/39)*cos(16*pi/39) + sin(pi/39)*sin(16*pi/39) - i*cos(pi/39)*sin(16*pi/39) - i*cos(16*pi/39)*sin(pi/39)))*(x - cos(pi/39)*cos(16*pi/39) - sin(pi/39)*sin(16*pi/39) - i*cos(16*pi/39)*sin(pi/39) + i*cos(pi/39)*sin(16*pi/39)))*(x - cos(pi/39)*cos(6*pi/13) + sin(pi/39)*sin(6*pi/13) - i*cos(pi/39)*sin(6*pi/13) - i*cos(6*pi/13)*sin(pi/39)))*(x - cos(pi/39)*cos(6*pi/13) - sin(pi/39)*sin(6*pi/13) - i*cos(6*pi/13)*sin(pi/39) + i*cos(pi/39)*sin(6*pi/13)))*(x + cos(pi/39)*cos(19*pi/39) + sin(pi/39)*sin(19*pi/39) - i*cos(pi/39)*sin(19*pi/39) + i*cos(19*pi/39)*sin(pi/39)))*(x + cos(pi/13)*cos(pi/39) + sin(pi/13)*sin(pi/39) - i*cos(pi/39)*sin(pi/13) + i*cos(pi/13)*sin(pi/39)))*(x - sin(pi/13)*sin(pi/39) + cos(pi/13)*cos(pi/39) + i*cos(pi/13)*sin(pi/39) + i*cos(pi/39)*sin(pi/13)))*(x + cos(pi/39)*cos(5*pi/39) + sin(pi/39)*sin(5*pi/39) - i*cos(pi/39)*sin(5*pi/39) + i*cos(5*pi/39)*sin(pi/39)))*(x - sin(pi/39)*sin(5*pi/39) + cos(pi/39)*cos(5*pi/39) + i*cos(pi/39)*sin(5*pi/39) + i*cos(5*pi/39)*sin(pi/39)))*(x + cos(pi/39)*cos(7*pi/39) + sin(pi/39)*sin(7*pi/39) - i*cos(pi/39)*sin(7*pi/39) + i*cos(7*pi/39)*sin(pi/39)))*(x - sin(pi/39)*sin(7*pi/39) + cos(pi/39)*cos(7*pi/39) + i*cos(pi/39)*sin(7*pi/39) + i*cos(7*pi/39)*sin(pi/39)))*(x + cos(pi/39)*cos(3*pi/13) + sin(pi/39)*sin(3*pi/13) - i*cos(pi/39)*sin(3*pi/13) + i*cos(3*pi/13)*sin(pi/39)))*(x - sin(pi/39)*sin(3*pi/13) + cos(pi/39)*cos(3*pi/13) + i*cos(pi/39)*sin(3*pi/13) + i*cos(3*pi/13)*sin(pi/39)))*(x + cos(pi/39)*cos(11*pi/39) + sin(pi/39)*sin(11*pi/39) - i*cos(pi/39)*sin(11*pi/39) + i*cos(11*pi/39)*sin(pi/39)))*(x - sin(pi/39)*sin(11*pi/39) + cos(pi/39)*cos(11*pi/39) + i*cos(pi/39)*sin(11*pi/39) + i*cos(11*pi/39)*sin(pi/39)))*(x + cos(pi/39)*cos(5*pi/13) + sin(pi/39)*sin(5*pi/13) - i*cos(pi/39)*sin(5*pi/13) + i*cos(5*pi/13)*sin(pi/39)))*(x - sin(pi/39)*sin(5*pi/13) + cos(pi/39)*cos(5*pi/13) + i*cos(pi/39)*sin(5*pi/13) + i*cos(5*pi/13)*sin(pi/39)))*(x + cos(pi/39)*cos(17*pi/39) + sin(pi/39)*sin(17*pi/39) - i*cos(pi/39)*sin(17*pi/39) + i*cos(17*pi/39)*sin(pi/39)))*(x - sin(pi/39)*sin(17*pi/39) + cos(pi/39)*cos(17*pi/39) + i*cos(pi/39)*sin(17*pi/39) + i*cos(17*pi/39)*sin(pi/39)))*(x - sin(pi/39)*sin(19*pi/39) + cos(pi/39)*cos(19*pi/39) + i*cos(pi/39)*sin(19*pi/39) + i*cos(19*pi/39)*sin(pi/39)))*(x + cos(pi/39)/2 + i*sin(pi/39)/2 + sqrt(3)*sin(pi/39)/2 - i*sqrt(3)*cos(pi/39)/2))*(x + cos(pi/39)/2 - sqrt(3)*sin(pi/39)/2 + i*sin(pi/39)/2 + i*sqrt(3)*cos(pi/39)/2)
Numerical answer [src] 
1.0 + x^39
1.0 + x^39
Combinatorics [src] 
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(1 + x)*\1 + x  - x/*\1 + x  + x  + x  + x  + x   + x   - x - x  - x  - x  - x  - x  /*\1 + x + x  + x  + x   + x   + x   + x   + x   - x  - x  - x  - x   - x   - x   - x   - x  /
(x+1)(x2−x+1)(x12−x11+x10−x9+x8−x7+x6−x5+x4−x3+x2−x+1)(x24+x23−x21−x20+x18+x17−x15−x14+x12−x10−x9+x7+x6−x4−x3+x+1)\left(x + 1\right) \left(x^{2} - x + 1\right) \left(x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\right) \left(x^{24} + x^{23} - x^{21} - x^{20} + x^{18} + x^{17} - x^{15} - x^{14} + x^{12} - x^{10} - x^{9} + x^{7} + x^{6} - x^{4} - x^{3} + x + 1\right) 
(1 + x)*(1 + x^2 - x)*(1 + x^2 + x^4 + x^6 + x^8 + x^10 + x^12 - x - x^3 - x^5 - x^7 - x^9 - x^11)*(1 + x + x^6 + x^7 + x^12 + x^17 + x^18 + x^23 + x^24 - x^3 - x^4 - x^9 - x^10 - x^14 - x^15 - x^20 - x^21)
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