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