Mister Exam
Factor polynomial x^8+y^8
The solution
Factorization
[src]
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| | / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ || | | / ___ / ___ / ___ / ___ ||
/ / ___________ ___________\\ / / ___________ ___________\\ / / ___________ ___________\\ / / ___________ ___________\\ | | ___ / 1 \/ 2 ___ / 1 \/ 2 ___ / 1 \/ 2 ___ / 1 \/ 2 || | | ___ / 1 \/ 2 ___ / 1 \/ 2 ___ / 1 \/ 2 ___ / 1 \/ 2 || | | ___ / 1 \/ 2 ___ / 1 \/ 2 ___ / 1 \/ 2 ___ / 1 \/ 2 || | | ___ / 1 \/ 2 ___ / 1 \/ 2 ___ / 1 \/ 2 ___ / 1 \/ 2 ||
| | / ___ / ___ || | | / ___ / ___ || | | / ___ / ___ || | | / ___ / ___ || | |\/ 2 * / - + ----- \/ 2 * / - - ----- I*\/ 2 * / - - ----- I*\/ 2 * / - + ----- || | |\/ 2 * / - - ----- \/ 2 * / - + ----- I*\/ 2 * / - - ----- I*\/ 2 * / - + ----- || | | \/ 2 * / - - ----- \/ 2 * / - + ----- I*\/ 2 * / - + ----- I*\/ 2 * / - - ----- || | |\/ 2 * / - - ----- \/ 2 * / - + ----- I*\/ 2 * / - - ----- I*\/ 2 * / - + ----- ||
| | / 1 \/ 2 / 1 \/ 2 || | | / 1 \/ 2 / 1 \/ 2 || | | / 1 \/ 2 / 1 \/ 2 || | | / 1 \/ 2 / 1 \/ 2 || | | \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 || | | \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 || | | \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 || | | \/ 2 4 \/ 2 4 \/ 2 4 \/ 2 4 ||
|x - y*|- / - - ----- + I* / - + ----- ||*|x - y*| / - - ----- - I* / - + ----- ||*|x - y*|- / - + ----- - I* / - - ----- ||*|x - y*| / - + ----- + I* / - - ----- ||*|x - y*|---------------------- - ---------------------- + ------------------------ + ------------------------||*|x - y*|---------------------- + ---------------------- + ------------------------ - ------------------------||*|x - y*|- ---------------------- - ---------------------- + ------------------------ - ------------------------||*|x - y*|---------------------- - ---------------------- - ------------------------ - ------------------------||
\ \ \/ 2 4 \/ 2 4 // \ \\/ 2 4 \/ 2 4 // \ \ \/ 2 4 \/ 2 4 // \ \\/ 2 4 \/ 2 4 // \ \ 2 2 2 2 // \ \ 2 2 2 2 // \ \ 2 2 2 2 // \ \ 2 2 2 2 //
$$\left(x - y \left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right) \left(x - y \left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right) \left(x - y \left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) \left(x - y \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) \left(x - y \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(x - y \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(x - y \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(x - y \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)$$
(((((((x - y*(-sqrt(1/2 - sqrt(2)/4) + i*sqrt(1/2 + sqrt(2)/4)))*(x - y*(sqrt(1/2 - sqrt(2)/4) - i*sqrt(1/2 + sqrt(2)/4))))*(x - y*(-sqrt(1/2 + sqrt(2)/4) - i*sqrt(1/2 - sqrt(2)/4))))*(x - y*(sqrt(1/2 + sqrt(2)/4) + i*sqrt(1/2 - sqrt(2)/4))))*(x - y*(sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 - sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 + i*sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 + i*sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2)))*(x - y*(sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 + sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 + i*sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 - i*sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2)))*(x - y*(-sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 - sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 + i*sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 - i*sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2)))*(x - y*(sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 - sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2 - i*sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 - i*sqrt(2)*sqrt(1/2 + sqrt(2)/4)/2))