Господин Экзамен
Разложить многочлен на множители 243*x^5-1
Решение
Разложение на множители
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/ ___________\ / ___________\ / ___________\ / ___________\
| / ___ | | / ___ | | / ___ | | / ___ |
| / 5 \/ 5 | | / 5 \/ 5 | | / 5 \/ 5 | | / 5 \/ 5 |
| ___ I* / - + ----- | | ___ I* / - + ----- | | ___ I* / - - ----- | | ___ I* / - - ----- |
| 1 \/ 5 \/ 8 8 | | 1 \/ 5 \/ 8 8 | | 1 \/ 5 \/ 8 8 | | 1 \/ 5 \/ 8 8 |
1*(x - 1/3)*|x + -- - ----- + ------------------|*|x + -- - ----- - ------------------|*|x + -- + ----- + ------------------|*|x + -- + ----- - ------------------|
\ 12 12 3 / \ 12 12 3 / \ 12 12 3 / \ 12 12 3 /
$$1 \left(x - \frac{1}{3}\right) \left(x + \left(- \frac{\sqrt{5}}{12} + \frac{1}{12} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3}\right)\right) \left(x - \left(- \frac{1}{12} + \frac{\sqrt{5}}{12} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{3}\right)\right) \left(x + \left(\frac{1}{12} + \frac{\sqrt{5}}{12} + \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{3}\right)\right) \left(x + \left(\frac{1}{12} + \frac{\sqrt{5}}{12} - \frac{i \sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}}{3}\right)\right)$$
((((1*(x - 1/3))*(x + (1/12 - sqrt(5)/12 + i*sqrt(5/8 + sqrt(5)/8)/3)))*(x + (1/12 - sqrt(5)/12 - i*sqrt(5/8 + sqrt(5)/8)/3)))*(x + (1/12 + sqrt(5)/12 + i*sqrt(5/8 - sqrt(5)/8)/3)))*(x + (1/12 + sqrt(5)/12 - i*sqrt(5/8 - sqrt(5)/8)/3))
Комбинаторика
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/ 2 3 4\ (-1 + 3*x)*\1 + 3*x + 9*x + 27*x + 81*x /
$$\left(3 x - 1\right) \left(81 x^{4} + 27 x^{3} + 9 x^{2} + 3 x + 1\right)$$
(-1 + 3*x)*(1 + 3*x + 9*x^2 + 27*x^3 + 81*x^4)