Общий множитель x^6+6*x^5-28*x^3-12*x^2+24*x

Объединение рациональных выражений

  /       /        /       2        \\\
x*\24 + x*\-12 + x*\-28 + x *(6 + x)///

$$x \left(x \left(x \left(x^{2} \left(x + 6\right) - 28\right) - 12\right) + 24\right)$$

Разложение на множители

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          |                                                                                             /   56        /  719   I*\/ 2031                                       24                                                176            | |                    /   56        /  719   I*\/ 2031                                       24                                                176                                                                                     | |                                                                                             /   56                                     24                                          /  719   I*\/ 2031                176            | |                                                                                             /   56                                     24                                          /  719   I*\/ 2031                176            |
          |                                                                                            /    -- - 2*3 /   --- + ----------  + ---------------------------------------------------------------------- - ------------------------- | |                   /    -- - 2*3 /   --- + ----------  + ---------------------------------------------------------------------- - -------------------------                                                                          | |                                                                                            /    -- - ---------------------------------------------------------------------- - 2*3 /   --- + ----------  - ------------------------- | |                                                                                            /    -- - ---------------------------------------------------------------------- - 2*3 /   --- + ----------  - ------------------------- |
          |                  ____________________________________________________________             /     3      \/     27       3                   ____________________________________________________________          __________________ | |                  /     3      \/     27       3                   ____________________________________________________________          __________________              ____________________________________________________________| |                  ____________________________________________________________             /     3              ____________________________________________________________     \/     27       3                __________________ | |                  ____________________________________________________________             /     3              ____________________________________________________________     \/     27       3                __________________ |
          |                 /             __________________                                         /                                                /             __________________                                      /           ______  | |                 /                                                /             __________________                                      /           ______              /             __________________                             | |                 /             __________________                                         /                    /             __________________                                                                  /           ______  | |                 /             __________________                                         /                    /             __________________                                                                  /           ______  |
          |                /             /           ______                                         /                                                /             /           ______                                      /  719   I*\/ 2031   | |                /                                                /             /           ______                                      /  719   I*\/ 2031              /             /           ______                              | |                /             /           ______                                         /                    /             /           ______                                                                  /  719   I*\/ 2031   | |                /             /           ______                                         /                    /             /           ______                                                                  /  719   I*\/ 2031   |
          |               /   28        /  719   I*\/ 2031                176                      /                                                /   28        /  719   I*\/ 2031                176               9*3 /   --- + ----------  | |               /                                                /   28        /  719   I*\/ 2031                176               9*3 /   --- + ----------            /   28        /  719   I*\/ 2031                176            | |               /   28        /  719   I*\/ 2031                176                      /                    /   28        /  719   I*\/ 2031                176                                           9*3 /   --- + ----------  | |               /   28        /  719   I*\/ 2031                176                      /                    /   28        /  719   I*\/ 2031                176                                           9*3 /   --- + ----------  |
          |              /    -- + 2*3 /   --- + ----------  + -------------------------          /                                                /    -- + 2*3 /   --- + ----------  + -------------------------      \/     27       3       | |              /                                                /    -- + 2*3 /   --- + ----------  + -------------------------      \/     27       3                /    -- + 2*3 /   --- + ----------  + ------------------------- | |              /    -- + 2*3 /   --- + ----------  + -------------------------          /                    /    -- + 2*3 /   --- + ----------  + -------------------------                                  \/     27       3       | |              /    -- + 2*3 /   --- + ----------  + -------------------------          /                    /    -- + 2*3 /   --- + ----------  + -------------------------                                  \/     27       3       |
          |             /     3      \/     27       3                __________________         /                                                /     3      \/     27       3                __________________                              | |             /                                                /     3      \/     27       3                __________________                                      /     3      \/     27       3                __________________ | |             /     3      \/     27       3                __________________         /                    /     3      \/     27       3                __________________                                                          | |             /     3      \/     27       3                __________________         /                    /     3      \/     27       3                __________________                                                          |
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          |           /                                             /  719   I*\/ 2031         /                                                /                                             /  719   I*\/ 2031                                | |           /                                                /                                             /  719   I*\/ 2031                                      /                                             /  719   I*\/ 2031   | |           /                                             /  719   I*\/ 2031         /                    /                                             /  719   I*\/ 2031                                                            | |           /                                             /  719   I*\/ 2031         /                    /                                             /  719   I*\/ 2031                                                            |
          |          /                                         9*3 /   --- + ----------       /                                                /                                         9*3 /   --- + ----------                               | |          /                                                /                                         9*3 /   --- + ----------                                    /                                         9*3 /   --- + ----------  | |          /                                         9*3 /   --- + ----------       /                    /                                         9*3 /   --- + ----------                                                           | |          /                                         9*3 /   --- + ----------       /                    /                                         9*3 /   --- + ----------                                                           |
          |        \/                                            \/     27       3          \/                                               \/                                            \/     27       3                                    | |        \/                                               \/                                            \/     27       3                                       \/                                            \/     27       3       | |        \/                                            \/     27       3          \/                   \/                                            \/     27       3                                                                | |        \/                                            \/     27       3          \/                   \/                                            \/     27       3                                                                |
(x + 2)*x*|x + 1 + ---------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------|*|x + 1 - ---------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------|*|x + 1 - ---------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------|*|x + 1 - ---------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------|
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$$x \left(x + 2\right) \left(x + \left(1 + \frac{\sqrt{\frac{56}{3} - 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}} + \frac{24}{\sqrt{\frac{28}{3} + \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}} + 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}} - \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}}}{2} + \frac{\sqrt{\frac{28}{3} + \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}} + 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}}{2}\right)\right) \left(x + \left(1 - \frac{\sqrt{\frac{56}{3} - 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}} + \frac{24}{\sqrt{\frac{28}{3} + \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}} + 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}} - \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}}}{2} + \frac{\sqrt{\frac{28}{3} + \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}} + 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}}{2}\right)\right) \left(x + \left(1 + \frac{\sqrt{\frac{56}{3} - 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}} - \frac{24}{\sqrt{\frac{28}{3} + \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}} + 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}} - \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}}}{2} - \frac{\sqrt{\frac{28}{3} + \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}} + 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}}{2}\right)\right) \left(x + \left(1 - \frac{\sqrt{\frac{28}{3} + \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}} + 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}}{2} - \frac{\sqrt{\frac{56}{3} - 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}} - \frac{24}{\sqrt{\frac{28}{3} + \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}} + 2 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}} - \frac{176}{9 \sqrt[3]{\frac{719}{27} + \frac{\sqrt{2031} i}{3}}}}}{2}\right)\right)$$

Комбинаторика

          /      4             2      3\
x*(2 + x)*\12 + x  - 12*x - 8*x  + 4*x /

$$x \left(x + 2\right) \left(x^{4} + 4 x^{3} - 8 x^{2} - 12 x + 12\right)$$

Общее упрощение

  /      5       2             4\
x*\24 + x  - 28*x  - 12*x + 6*x /

$$x \left(x^{5} + 6 x^{4} - 28 x^{2} - 12 x + 24\right)$$

Собрать выражение

 6       3       2      5       
x  - 28*x  - 12*x  + 6*x  + 24*x

$$x^{6} + 6 x^{5} - 28 x^{3} - 12 x^{2} + 24 x$$