Mister Exam
Factor polynomial x^5+32
The solution
Factorization
[src]
/ ___________ ___________ ___________ ___________\ / ___________ ___________ ___________ ___________\
| / ___ / ___ / ___ / ___ | | / ___ / ___ / ___ / ___ |
/ ___________\ / ___________ ___________\ | ___________ ___________ / 5 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 | | / 5 \/ 5 ___________ ___________ / 5 \/ 5 ___ / 5 \/ 5 ___ / 5 \/ 5 |
| ___ / ___ | | ___ / ___ / ___ | | / ___ / ___ I* / - - ----- I* / - + ----- I*\/ 5 * / - - ----- I*\/ 5 * / - + ----- | | I* / - + ----- / ___ / ___ I* / - - ----- I*\/ 5 * / - - ----- I*\/ 5 * / - + ----- |
| 1 \/ 5 / 5 \/ 5 | | 1 \/ 5 / 5 \/ 5 ___ / 5 \/ 5 | | 1 / 5 \/ 5 / 5 \/ 5 \/ 8 8 \/ 8 8 \/ 8 8 \/ 8 8 | | 1 \/ 8 8 / 5 \/ 5 / 5 \/ 5 \/ 8 8 \/ 8 8 \/ 8 8 |
(x + 2)*|x + - - - ----- - 2*I* / - - ----- |*|x + - - + ----- + I* / - - ----- + I*\/ 5 * / - - ----- |*|x + - - - 2* / - - ----- * / - + ----- + ------------------ + ------------------ - ------------------------ + ------------------------|*|x + - - - ------------------ + 2* / - - ----- * / - + ----- + ------------------ - ------------------------ - ------------------------|
\ 2 2 \/ 8 8 / \ 2 2 \/ 8 8 \/ 8 8 / \ 2 \/ 8 8 \/ 8 8 2 2 2 2 / \ 2 2 \/ 8 8 \/ 8 8 2 2 2 /
$$\left(x + 2\right) \left(x + \left(- \frac{\sqrt{5}}{2} - \frac{1}{2} - 2 i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right) \left(x + \left(- \frac{1}{2} + \frac{\sqrt{5}}{2} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} + \sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right) \left(x + \left(- 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{1}{2} - \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right)\right) \left(x + \left(- \frac{1}{2} + 2 \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)\right)$$
((((x + 2)*(x - 1/2 - sqrt(5)/2 - 2*i*sqrt(5/8 - sqrt(5)/8)))*(x - 1/2 + sqrt(5)/2 + i*sqrt(5/8 - sqrt(5)/8) + i*sqrt(5)*sqrt(5/8 - sqrt(5)/8)))*(x - 1/2 - 2*sqrt(5/8 - sqrt(5)/8)*sqrt(5/8 + sqrt(5)/8) + i*sqrt(5/8 - sqrt(5)/8)/2 + i*sqrt(5/8 + sqrt(5)/8)/2 - i*sqrt(5)*sqrt(5/8 - sqrt(5)/8)/2 + i*sqrt(5)*sqrt(5/8 + sqrt(5)/8)/2))*(x - 1/2 - i*sqrt(5/8 + sqrt(5)/8)/2 + 2*sqrt(5/8 - sqrt(5)/8)*sqrt(5/8 + sqrt(5)/8) + i*sqrt(5/8 - sqrt(5)/8)/2 - i*sqrt(5)*sqrt(5/8 - sqrt(5)/8)/2 - i*sqrt(5)*sqrt(5/8 + sqrt(5)/8)/2)
Combinatorics
[src]
/ 4 3 2\ (2 + x)*\16 + x - 8*x - 2*x + 4*x /
$$\left(x + 2\right) \left(x^{4} - 2 x^{3} + 4 x^{2} - 8 x + 16\right)$$
(2 + x)*(16 + x^4 - 8*x - 2*x^3 + 4*x^2)