Господин Экзамен
Разложить многочлен на множители x^8+1
Решение
Разложение на множители
[src]
/ ___________ ___________ ___________ ___________\ / ___________ ___________ ___________ ___________\ / ___________ ___________ ___________ ___________\ / ___________ ___________ ___________ ___________\
| / ___ / ___ / ___ / ___ | | / ___ / ___ / ___ / ___ | | / ___ / ___ / ___ / ___ | | / ___ / ___ / ___ / ___ |
/ ___________ ___________\ / ___________ ___________\ / ___________ ___________\ / ___________ ___________\ | ___ / 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 |
1*|x + / - - ----- - I* / - + ----- |*|x + - / - - ----- + I* / - + ----- |*|x + / - + ----- + I* / - - ----- |*|x + - / - + ----- - I* / - - ----- |*|x + - ---------------------- + ---------------------- - ------------------------ - ------------------------|*|x + - ---------------------- - ---------------------- - ------------------------ + ------------------------|*|x + ---------------------- + ---------------------- - ------------------------ + ------------------------|*|x + - ---------------------- + ---------------------- + ------------------------ + ------------------------|
\ \/ 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 /
$$\left(x - \left(\sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right) 1 \left(x + \left(\sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right) \left(x + \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right) \left(x - \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right) \left(x - \left(- \frac{\sqrt{2} \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(x - \left(\frac{\sqrt{2} \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(x + \left(\frac{\sqrt{2} \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(x + \left(- \frac{\sqrt{2} \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{- \frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)$$
(((((((1*(x + (sqrt(1/2 - sqrt(2)/4) - i*sqrt(1/2 + sqrt(2)/4))))*(x - (sqrt(1/2 - sqrt(2)/4) + i*sqrt(1/2 + sqrt(2)/4))))*(x + (sqrt(1/2 + sqrt(2)/4) + i*sqrt(1/2 - sqrt(2)/4))))*(x - (sqrt(1/2 + sqrt(2)/4) - i*sqrt(1/2 - sqrt(2)/4))))*(x - (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 - (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 + (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 - (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))