Sr Examen
Factorizar el polinomio z^10*g^20-1
Solución
Factorización
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/ _____ _____ \ / _____ _____ \ / _____ _____ \ / _____ _____ \ / _____ _____\ / _____ _____\ / _____ _____\ / _____ _____\ / _____ _____\ / _____ _____\ / _____ _____\ / _____ _____\ / _____ _____ \ / _____ _____ \ / _____ _____ \ / _____ _____ \
| / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 | | / 1 ___ / 1 |
| / --- \/ 5 * / --- ___________| | / --- \/ 5 * / --- ___________| | / --- \/ 5 * / --- ___________| | / --- \/ 5 * / --- ___________| | ___________ I* / --- I*\/ 5 * / --- | | ___________ I* / --- I*\/ 5 * / --- | | ___________ I* / --- I*\/ 5 * / --- | | ___________ I* / --- I*\/ 5 * / --- | | ___________ I* / --- I*\/ 5 * / --- | | ___________ I* / --- I*\/ 5 * / --- | | ___________ I* / --- I*\/ 5 * / --- | | ___________ I* / --- I*\/ 5 * / --- | | / --- \/ 5 * / --- ___________| | / --- \/ 5 * / --- ___________| | / --- \/ 5 * / --- ___________| | / --- \/ 5 * / --- ___________|
/ _____\ / _____\ | 20/ 10 20/ 10 _____ / ___ | | 20/ 10 20/ 10 _____ / ___ | | 20/ 10 20/ 10 _____ / ___ | | 20/ 10 20/ 10 _____ / ___ | | _____ / ___ 20/ 10 20/ 10 | | _____ / ___ 20/ 10 20/ 10 | | _____ / ___ 20/ 10 20/ 10 | | _____ / ___ 20/ 10 20/ 10 | | _____ / ___ 20/ 10 20/ 10 | | _____ / ___ 20/ 10 20/ 10 | | _____ / ___ 20/ 10 20/ 10 | | _____ / ___ 20/ 10 20/ 10 | | 20/ 10 20/ 10 _____ / ___ | | 20/ 10 20/ 10 _____ / ___ | | 20/ 10 20/ 10 _____ / ___ | | 20/ 10 20/ 10 _____ / ___ | / _____\ / _____\
| / 1 | | / 1 | | \/ z \/ z / 1 / 5 \/ 5 | | \/ z \/ z / 1 / 5 \/ 5 | | \/ z \/ z / 1 / 5 \/ 5 | | \/ z \/ z / 1 / 5 \/ 5 | | / 1 / 5 \/ 5 \/ z \/ z | | / 1 / 5 \/ 5 \/ z \/ z | | / 1 / 5 \/ 5 \/ z \/ z | | / 1 / 5 \/ 5 \/ z \/ z | | / 1 / 5 \/ 5 \/ z \/ z | | / 1 / 5 \/ 5 \/ z \/ z | | / 1 / 5 \/ 5 \/ z \/ z | | / 1 / 5 \/ 5 \/ z \/ z | | \/ z \/ z / 1 / 5 \/ 5 | | \/ z \/ z / 1 / 5 \/ 5 | | \/ z \/ z / 1 / 5 \/ 5 | | \/ z \/ z / 1 / 5 \/ 5 | | / 1 | | / 1 |
|g + I* / --- |*|g - I* / --- |*|g + ---------- + ---------------- + I* / --- * / - - ----- |*|g + ---------- + ---------------- - I* / --- * / - - ----- |*|g + - ---------- + ---------------- + I* / --- * / - + ----- |*|g + - ---------- + ---------------- - I* / --- * / - + ----- |*|g + / --- * / - - ----- - ------------ - ------------------|*|g + / --- * / - - ----- + ------------ + ------------------|*|g + - / --- * / - - ----- - ------------ - ------------------|*|g + - / --- * / - - ----- + ------------ + ------------------|*|g + / --- * / - + ----- + ------------ - ------------------|*|g + / --- * / - + ----- - ------------ + ------------------|*|g + - / --- * / - + ----- + ------------ - ------------------|*|g + - / --- * / - + ----- - ------------ + ------------------|*|g + ---------- - ---------------- + I* / --- * / - + ----- |*|g + ---------- - ---------------- - I* / --- * / - + ----- |*|g + - ---------- - ---------------- + I* / --- * / - - ----- |*|g + - ---------- - ---------------- - I* / --- * / - - ----- |*|g + / --- |*|g - / --- |
| 20/ 10 | | 20/ 10 | | 4 4 20/ 10 \/ 8 8 | | 4 4 20/ 10 \/ 8 8 | | 4 4 20/ 10 \/ 8 8 | | 4 4 20/ 10 \/ 8 8 | | 20/ 10 \/ 8 8 4 4 | | 20/ 10 \/ 8 8 4 4 | | 20/ 10 \/ 8 8 4 4 | | 20/ 10 \/ 8 8 4 4 | | 20/ 10 \/ 8 8 4 4 | | 20/ 10 \/ 8 8 4 4 | | 20/ 10 \/ 8 8 4 4 | | 20/ 10 \/ 8 8 4 4 | | 4 4 20/ 10 \/ 8 8 | | 4 4 20/ 10 \/ 8 8 | | 4 4 20/ 10 \/ 8 8 | | 4 4 20/ 10 \/ 8 8 | | 20/ 10 | | 20/ 10 |
\ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z / \ \/ z /
$$\left(g - i \sqrt[20]{\frac{1}{z^{10}}}\right) \left(g + i \sqrt[20]{\frac{1}{z^{10}}}\right) \left(g + \left(\frac{\sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt{5} \sqrt[20]{\frac{1}{z^{10}}}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt[20]{\frac{1}{z^{10}}}\right)\right) \left(g + \left(\frac{\sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt{5} \sqrt[20]{\frac{1}{z^{10}}}}{4} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt[20]{\frac{1}{z^{10}}}\right)\right) \left(g + \left(- \frac{\sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt{5} \sqrt[20]{\frac{1}{z^{10}}}}{4} + i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt[20]{\frac{1}{z^{10}}}\right)\right) \left(g + \left(- \frac{\sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt{5} \sqrt[20]{\frac{1}{z^{10}}}}{4} - i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt[20]{\frac{1}{z^{10}}}\right)\right) \left(g + \left(\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt[20]{\frac{1}{z^{10}}} - \frac{\sqrt{5} i \sqrt[20]{\frac{1}{z^{10}}}}{4} - \frac{i \sqrt[20]{\frac{1}{z^{10}}}}{4}\right)\right) \left(g + \left(\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt[20]{\frac{1}{z^{10}}} + \frac{i \sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt{5} i \sqrt[20]{\frac{1}{z^{10}}}}{4}\right)\right) \left(g + \left(- \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt[20]{\frac{1}{z^{10}}} - \frac{\sqrt{5} i \sqrt[20]{\frac{1}{z^{10}}}}{4} - \frac{i \sqrt[20]{\frac{1}{z^{10}}}}{4}\right)\right) \left(g + \left(- \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt[20]{\frac{1}{z^{10}}} + \frac{i \sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt{5} i \sqrt[20]{\frac{1}{z^{10}}}}{4}\right)\right) \left(g + \left(\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt[20]{\frac{1}{z^{10}}} - \frac{\sqrt{5} i \sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{i \sqrt[20]{\frac{1}{z^{10}}}}{4}\right)\right) \left(g + \left(\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt[20]{\frac{1}{z^{10}}} - \frac{i \sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt{5} i \sqrt[20]{\frac{1}{z^{10}}}}{4}\right)\right) \left(g + \left(- \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt[20]{\frac{1}{z^{10}}} - \frac{\sqrt{5} i \sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{i \sqrt[20]{\frac{1}{z^{10}}}}{4}\right)\right) \left(g + \left(- \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt[20]{\frac{1}{z^{10}}} - \frac{i \sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt{5} i \sqrt[20]{\frac{1}{z^{10}}}}{4}\right)\right) \left(g + \left(- \frac{\sqrt{5} \sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt[20]{\frac{1}{z^{10}}}}{4} + i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt[20]{\frac{1}{z^{10}}}\right)\right) \left(g + \left(- \frac{\sqrt{5} \sqrt[20]{\frac{1}{z^{10}}}}{4} + \frac{\sqrt[20]{\frac{1}{z^{10}}}}{4} - i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \sqrt[20]{\frac{1}{z^{10}}}\right)\right) \left(g + \left(- \frac{\sqrt{5} \sqrt[20]{\frac{1}{z^{10}}}}{4} - \frac{\sqrt[20]{\frac{1}{z^{10}}}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt[20]{\frac{1}{z^{10}}}\right)\right) \left(g + \left(- \frac{\sqrt{5} \sqrt[20]{\frac{1}{z^{10}}}}{4} - \frac{\sqrt[20]{\frac{1}{z^{10}}}}{4} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt[20]{\frac{1}{z^{10}}}\right)\right) \left(g + \sqrt[20]{\frac{1}{z^{10}}}\right) \left(g - \sqrt[20]{\frac{1}{z^{10}}}\right)$$
(((((((((((((((((((g + i*(z^(-10))^(1/20))*(g - i*(z^(-10))^(1/20)))*(g + (z^(-10))^(1/20)/4 + sqrt(5)*(z^(-10))^(1/20)/4 + i*(z^(-10))^(1/20)*sqrt(5/8 - sqrt(5)/8)))*(g + (z^(-10))^(1/20)/4 + sqrt(5)*(z^(-10))^(1/20)/4 - i*(z^(-10))^(1/20)*sqrt(5/8 - sqrt(5)/8)))*(g - (z^(-10))^(1/20)/4 + sqrt(5)*(z^(-10))^(1/20)/4 + i*(z^(-10))^(1/20)*sqrt(5/8 + sqrt(5)/8)))*(g - (z^(-10))^(1/20)/4 + sqrt(5)*(z^(-10))^(1/20)/4 - i*(z^(-10))^(1/20)*sqrt(5/8 + sqrt(5)/8)))*(g + (z^(-10))^(1/20)*sqrt(5/8 - sqrt(5)/8) - i*(z^(-10))^(1/20)/4 - i*sqrt(5)*(z^(-10))^(1/20)/4))*(g + (z^(-10))^(1/20)*sqrt(5/8 - sqrt(5)/8) + i*(z^(-10))^(1/20)/4 + i*sqrt(5)*(z^(-10))^(1/20)/4))*(g - (z^(-10))^(1/20)*sqrt(5/8 - sqrt(5)/8) - i*(z^(-10))^(1/20)/4 - i*sqrt(5)*(z^(-10))^(1/20)/4))*(g - (z^(-10))^(1/20)*sqrt(5/8 - sqrt(5)/8) + i*(z^(-10))^(1/20)/4 + i*sqrt(5)*(z^(-10))^(1/20)/4))*(g + (z^(-10))^(1/20)*sqrt(5/8 + sqrt(5)/8) + i*(z^(-10))^(1/20)/4 - i*sqrt(5)*(z^(-10))^(1/20)/4))*(g + (z^(-10))^(1/20)*sqrt(5/8 + sqrt(5)/8) - i*(z^(-10))^(1/20)/4 + i*sqrt(5)*(z^(-10))^(1/20)/4))*(g - (z^(-10))^(1/20)*sqrt(5/8 + sqrt(5)/8) + i*(z^(-10))^(1/20)/4 - i*sqrt(5)*(z^(-10))^(1/20)/4))*(g - (z^(-10))^(1/20)*sqrt(5/8 + sqrt(5)/8) - i*(z^(-10))^(1/20)/4 + i*sqrt(5)*(z^(-10))^(1/20)/4))*(g + (z^(-10))^(1/20)/4 - sqrt(5)*(z^(-10))^(1/20)/4 + i*(z^(-10))^(1/20)*sqrt(5/8 + sqrt(5)/8)))*(g + (z^(-10))^(1/20)/4 - sqrt(5)*(z^(-10))^(1/20)/4 - i*(z^(-10))^(1/20)*sqrt(5/8 + sqrt(5)/8)))*(g - (z^(-10))^(1/20)/4 - sqrt(5)*(z^(-10))^(1/20)/4 + i*(z^(-10))^(1/20)*sqrt(5/8 - sqrt(5)/8)))*(g - (z^(-10))^(1/20)/4 - sqrt(5)*(z^(-10))^(1/20)/4 - i*(z^(-10))^(1/20)*sqrt(5/8 - sqrt(5)/8)))*(g + (z^(-10))^(1/20)))*(g - (z^(-10))^(1/20))
Combinatoria
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/ 2\ / 2\ / 2 4 2 6 3 8 4\ / 4 2 8 4 2 6 3\ \1 + z*g /*\-1 + z*g /*\1 + z*g + g *z + g *z + g *z /*\1 + g *z + g *z - z*g - g *z /
$$\left(g^{2} z - 1\right) \left(g^{2} z + 1\right) \left(g^{8} z^{4} - g^{6} z^{3} + g^{4} z^{2} - g^{2} z + 1\right) \left(g^{8} z^{4} + g^{6} z^{3} + g^{4} z^{2} + g^{2} z + 1\right)$$
(1 + z*g^2)*(-1 + z*g^2)*(1 + z*g^2 + g^4*z^2 + g^6*z^3 + g^8*z^4)*(1 + g^4*z^2 + g^8*z^4 - z*g^2 - g^6*z^3)