Господин Экзамен
Разложить многочлен на множители a^8-b^22
Решение
Разложение на множители
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/ _____ _____\ / _____ _____\ / _____ _____\ / _____ _____\
/ _____\ / _____\ | ___ 8 / 22 ___ 8 / 22 | | ___ 8 / 22 ___ 8 / 22 | | ___ 8 / 22 ___ 8 / 22 | | ___ 8 / 22 ___ 8 / 22 | / _____\ / _____\
| 8 / 22 | | 8 / 22 | | \/ 2 *\/ b I*\/ 2 *\/ b | | \/ 2 *\/ b I*\/ 2 *\/ b | | \/ 2 *\/ b I*\/ 2 *\/ b | | \/ 2 *\/ b I*\/ 2 *\/ b | | 8 / 22 | | 8 / 22 |
1*\a + I*\/ b /*\a - I*\/ b /*|a + -------------- + ----------------|*|a + -------------- - ----------------|*|a + - -------------- + ----------------|*|a + - -------------- - ----------------|*\a + \/ b /*\a - \/ b /
\ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$\left(a - i \sqrt[8]{b^{22}}\right) 1 \left(a + i \sqrt[8]{b^{22}}\right) \left(a + \left(\frac{\sqrt{2} \sqrt[8]{b^{22}}}{2} + \frac{\sqrt{2} i \sqrt[8]{b^{22}}}{2}\right)\right) \left(a + \left(\frac{\sqrt{2} \sqrt[8]{b^{22}}}{2} - \frac{\sqrt{2} i \sqrt[8]{b^{22}}}{2}\right)\right) \left(a - \left(\frac{\sqrt{2} \sqrt[8]{b^{22}}}{2} - \frac{\sqrt{2} i \sqrt[8]{b^{22}}}{2}\right)\right) \left(a - \left(\frac{\sqrt{2} \sqrt[8]{b^{22}}}{2} + \frac{\sqrt{2} i \sqrt[8]{b^{22}}}{2}\right)\right) \left(a + \sqrt[8]{b^{22}}\right) \left(a - \sqrt[8]{b^{22}}\right)$$
(((((((1*(a + i*(b^22)^(1/8)))*(a - i*(b^22)^(1/8)))*(a + (sqrt(2)*(b^22)^(1/8)/2 + i*sqrt(2)*(b^22)^(1/8)/2)))*(a + (sqrt(2)*(b^22)^(1/8)/2 - i*sqrt(2)*(b^22)^(1/8)/2)))*(a - (sqrt(2)*(b^22)^(1/8)/2 + i*sqrt(2)*(b^22)^(1/8)/2)))*(a - (sqrt(2)*(b^22)^(1/8)/2 - i*sqrt(2)*(b^22)^(1/8)/2)))*(a + (b^22)^(1/8)))*(a - (b^22)^(1/8))
Комбинаторика
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/ 4 11\ / 4 11\ \a + b /*\a - b /
$$\left(- b^{11} + a^{4}\right) \left(b^{11} + a^{4}\right)$$
(a^4 + b^11)*(a^4 - b^11)