Господин Экзамен
Разложить многочлен на множители a^9+27*b^3
Решение
Разложение на множители
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/ _____ _____\ / _____ _____\
/ _____\ | 3 ___ 9 / 3 5/6 9 / 3 | | 3 ___ 9 / 3 5/6 9 / 3 | / _____ _____ \ / _____ _____ \ / _____ _____ \ / _____ _____ \ / _____ _____ \ / _____ _____ \
| 3 ___ 9 / 3 | | \/ 3 *\/ -b I*3 *\/ -b | | \/ 3 *\/ -b I*3 *\/ -b | | 3 ___ 9 / 3 /pi\ 3 ___ 9 / 3 /pi\| | 3 ___ 9 / 3 /pi\ 3 ___ 9 / 3 /pi\| | 3 ___ 9 / 3 /2*pi\ 3 ___ 9 / 3 /2*pi\| | 3 ___ 9 / 3 /2*pi\ 3 ___ 9 / 3 /2*pi\| | 3 ___ 9 / 3 /4*pi\ 3 ___ 9 / 3 /4*pi\| | 3 ___ 9 / 3 /4*pi\ 3 ___ 9 / 3 /4*pi\|
1*\a - \/ 3 *\/ -b /*|a + -------------- + ---------------|*|a + -------------- - ---------------|*|a + \/ 3 *\/ -b *cos|--| + I*\/ 3 *\/ -b *sin|--||*|a + \/ 3 *\/ -b *cos|--| - I*\/ 3 *\/ -b *sin|--||*|a + - \/ 3 *\/ -b *cos|----| + I*\/ 3 *\/ -b *sin|----||*|a + - \/ 3 *\/ -b *cos|----| - I*\/ 3 *\/ -b *sin|----||*|a + - \/ 3 *\/ -b *cos|----| + I*\/ 3 *\/ -b *sin|----||*|a + - \/ 3 *\/ -b *cos|----| - I*\/ 3 *\/ -b *sin|----||
\ 2 2 / \ 2 2 / \ \9 / \9 // \ \9 / \9 // \ \ 9 / \ 9 // \ \ 9 / \ 9 // \ \ 9 / \ 9 // \ \ 9 / \ 9 //
$$1 \left(a - \sqrt[3]{3} \sqrt[9]{- b^{3}}\right) \left(a + \left(\frac{\sqrt[3]{3} \sqrt[9]{- b^{3}}}{2} + \frac{3^{\frac{5}{6}} i \sqrt[9]{- b^{3}}}{2}\right)\right) \left(a + \left(\frac{\sqrt[3]{3} \sqrt[9]{- b^{3}}}{2} - \frac{3^{\frac{5}{6}} i \sqrt[9]{- b^{3}}}{2}\right)\right) \left(a + \left(\sqrt[3]{3} \sqrt[9]{- b^{3}} \cos{\left(\frac{\pi}{9} \right)} + \sqrt[3]{3} i \sqrt[9]{- b^{3}} \sin{\left(\frac{\pi}{9} \right)}\right)\right) \left(a + \left(\sqrt[3]{3} \sqrt[9]{- b^{3}} \cos{\left(\frac{\pi}{9} \right)} - \sqrt[3]{3} i \sqrt[9]{- b^{3}} \sin{\left(\frac{\pi}{9} \right)}\right)\right) \left(a - \left(\sqrt[3]{3} \sqrt[9]{- b^{3}} \cos{\left(\frac{2 \pi}{9} \right)} - \sqrt[3]{3} i \sqrt[9]{- b^{3}} \sin{\left(\frac{2 \pi}{9} \right)}\right)\right) \left(a - \left(\sqrt[3]{3} \sqrt[9]{- b^{3}} \cos{\left(\frac{2 \pi}{9} \right)} + \sqrt[3]{3} i \sqrt[9]{- b^{3}} \sin{\left(\frac{2 \pi}{9} \right)}\right)\right) \left(a - \left(\sqrt[3]{3} \sqrt[9]{- b^{3}} \cos{\left(\frac{4 \pi}{9} \right)} - \sqrt[3]{3} i \sqrt[9]{- b^{3}} \sin{\left(\frac{4 \pi}{9} \right)}\right)\right) \left(a - \left(\sqrt[3]{3} \sqrt[9]{- b^{3}} \cos{\left(\frac{4 \pi}{9} \right)} + \sqrt[3]{3} i \sqrt[9]{- b^{3}} \sin{\left(\frac{4 \pi}{9} \right)}\right)\right)$$
((((((((1*(a - 3^(1/3)*(-b^3)^(1/9)))*(a + (3^(1/3)*(-b^3)^(1/9)/2 + i*3^(5/6)*(-b^3)^(1/9)/2)))*(a + (3^(1/3)*(-b^3)^(1/9)/2 - i*3^(5/6)*(-b^3)^(1/9)/2)))*(a + (3^(1/3)*(-b^3)^(1/9)*cos(pi/9) + i*3^(1/3)*(-b^3)^(1/9)*sin(pi/9))))*(a + (3^(1/3)*(-b^3)^(1/9)*cos(pi/9) - i*3^(1/3)*(-b^3)^(1/9)*sin(pi/9))))*(a - (3^(1/3)*(-b^3)^(1/9)*cos(2*pi/9) + i*3^(1/3)*(-b^3)^(1/9)*sin(2*pi/9))))*(a - (3^(1/3)*(-b^3)^(1/9)*cos(2*pi/9) - i*3^(1/3)*(-b^3)^(1/9)*sin(2*pi/9))))*(a - (3^(1/3)*(-b^3)^(1/9)*cos(4*pi/9) + i*3^(1/3)*(-b^3)^(1/9)*sin(4*pi/9))))*(a - (3^(1/3)*(-b^3)^(1/9)*cos(4*pi/9) - i*3^(1/3)*(-b^3)^(1/9)*sin(4*pi/9)))
Комбинаторика
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/ 3 \ / 6 2 3\ \a + 3*b/*\a + 9*b - 3*b*a /
$$\left(a^{3} + 3 b\right) \left(a^{6} - 3 a^{3} b + 9 b^{2}\right)$$
(a^3 + 3*b)*(a^6 + 9*b^2 - 3*b*a^3)