Mister Exam
Factor polynomial x^8-y^8
The solution
Factorization
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/ / ___ ___\\ / / ___ ___\\ / / ___ ___\\ / / ___ ___\\
| | \/ 2 I*\/ 2 || | | \/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 || | |\/ 2 I*\/ 2 ||
(x + y)*(x - y)*(x + I*y)*(x - I*y)*|x - y*|- ----- - -------||*|x - y*|- ----- + -------||*|x - y*|----- - -------||*|x - y*|----- + -------||
\ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 //
$$\left(x - y\right) \left(x + y\right) \left(x + i y\right) \left(x - i y\right) \left(x - y \left(- \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right)\right) \left(x - y \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)\right) \left(x - y \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right)\right) \left(x - y \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)\right)$$
(((((((x + y)*(x - y))*(x + i*y))*(x - i*y))*(x - y*(-sqrt(2)/2 - i*sqrt(2)/2)))*(x - y*(-sqrt(2)/2 + i*sqrt(2)/2)))*(x - y*(sqrt(2)/2 - i*sqrt(2)/2)))*(x - y*(sqrt(2)/2 + i*sqrt(2)/2))
Combinatorics
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/ 2 2\ / 4 4\ (x + y)*(x - y)*\x + y /*\x + y /
$$\left(x - y\right) \left(x + y\right) \left(x^{2} + y^{2}\right) \left(x^{4} + y^{4}\right)$$
(x + y)*(x - y)*(x^2 + y^2)*(x^4 + y^4)