Общий множитель a^5+b^5

Комбинаторика

        / 4    4    2  2      3      3\
(a + b)*\a  + b  + a *b  - a*b  - b*a /

$$\left(a + b\right) \left(a^{4} - a^{3} b + a^{2} b^{2} - a b^{3} + b^{4}\right)$$

Разложение на множители

                                                 /      /                   ___________                ___________\\ /      /                                               ___________          ___________                ___________                ___________\\ /      /                                               ___________          ___________                ___________                ___________\\
                                                 |      |                  /       ___                /       ___ || |      |                                              /       ___          /       ___                /       ___                /       ___ || |      |                                              /       ___          /       ___                /       ___                /       ___ ||
        /      /                   ___________\\ |      |                 /  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            //

$$\left(a + b\right) \left(a - b \left(\frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right)\right) \left(a - b \left(- \frac{\sqrt{5}}{4} + \frac{1}{4} - \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)\right) \left(a - b \left(\frac{1}{4} + \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}\right)\right) \left(a - b \left(- \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right)\right)$$