Разложить многочлен на множители 4*x^4+12*x^3+13*x^2+8*x+1

Комбинаторика

       4             3       2
1 + 4*x  + 8*x + 12*x  + 13*x 

$$4 x^{4} + 12 x^{3} + 13 x^{2} + 8 x + 1$$

Объединение рациональных выражений

1 + x*(8 + x*(13 + 4*x*(3 + x)))

$$x \left(x \left(4 x \left(x + 3\right) + 13\right) + 8\right) + 1$$

Разложение на множители

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|                    /   1                                     1                                           /   109    \/ 642                 71                                                                                      | |                                                                                             /   1                                     1                                           /   109    \/ 642                 71             | |                                                                                             /   1                                     1                                           /   109    \/ 642                 71             | |                                                                                             /   1                                     1                                           /   109    \/ 642                 71             |
|                   /    - + ---------------------------------------------------------------------- - 2*3 /   ----- + -------  + --------------------------                                                                          | |                                                                                            /    - - ---------------------------------------------------------------------- - 2*3 /   ----- + -------  + -------------------------- | |                                                                                            /    - - ---------------------------------------------------------------------- - 2*3 /   ----- + -------  + -------------------------- | |                                                                                            /    - + ---------------------------------------------------------------------- - 2*3 /   ----- + -------  + -------------------------- |
|                  /     6             ____________________________________________________________     \/    13824     576               _________________              ____________________________________________________________| |                  ____________________________________________________________             /     6             ____________________________________________________________     \/    13824     576               _________________ | |                  ____________________________________________________________             /     6             ____________________________________________________________     \/    13824     576               _________________ | |                  ____________________________________________________________             /     6             ____________________________________________________________     \/    13824     576               _________________ |
|                 /                   /             _________________                                                                    /           _____              /             _________________                              | |                 /             _________________                                          /                   /             _________________                                                                    /           _____  | |                 /             _________________                                          /                   /             _________________                                                                    /           _____  | |                 /             _________________                                          /                   /             _________________                                                                    /           _____  |
|                /                   /             /           _____                                                                    /   109    \/ 642              /             /           _____                               | |                /             /           _____                                          /                   /             /           _____                                                                    /   109    \/ 642   | |                /             /           _____                                          /                   /             /           _____                                                                    /   109    \/ 642   | |                /             /           _____                                          /                   /             /           _____                                                                    /   109    \/ 642   |
|               /                   /   1         /   109    \/ 642                 71                                           288*3 /   ----- + -------            /   1         /   109    \/ 642                 71             | |               /   1         /   109    \/ 642                 71                       /                   /   1         /   109    \/ 642                 71                                           288*3 /   ----- + -------  | |               /   1         /   109    \/ 642                 71                       /                   /   1         /   109    \/ 642                 71                                           288*3 /   ----- + -------  | |               /   1         /   109    \/ 642                 71                       /                   /   1         /   109    \/ 642                 71                                           288*3 /   ----- + -------  |
|              /                   /    -- + 2*3 /   ----- + -------  - --------------------------                                   \/    13824     576             /    -- + 2*3 /   ----- + -------  - -------------------------- | |              /    -- + 2*3 /   ----- + -------  - --------------------------          /                   /    -- + 2*3 /   ----- + -------  - --------------------------                                   \/    13824     576    | |              /    -- + 2*3 /   ----- + -------  - --------------------------          /                   /    -- + 2*3 /   ----- + -------  - --------------------------                                   \/    13824     576    | |              /    -- + 2*3 /   ----- + -------  - --------------------------          /                   /    -- + 2*3 /   ----- + -------  - --------------------------                                   \/    13824     576    |
|             /                   /     12     \/    13824     576               _________________                                                                  /     12     \/    13824     576               _________________ | |             /     12     \/    13824     576               _________________         /                   /     12     \/    13824     576               _________________                                                          | |             /     12     \/    13824     576               _________________         /                   /     12     \/    13824     576               _________________                                                          | |             /     12     \/    13824     576               _________________         /                   /     12     \/    13824     576               _________________                                                          |
|            /                   /                                              /           _____                                                                  /                                              /           _____  | |            /                                              /           _____         /                   /                                              /           _____                                                           | |            /                                              /           _____         /                   /                                              /           _____                                                           | |            /                                              /           _____         /                   /                                              /           _____                                                           |
|           /                   /                                              /   109    \/ 642                                                                  /                                              /   109    \/ 642   | |           /                                              /   109    \/ 642         /                   /                                              /   109    \/ 642                                                            | |           /                                              /   109    \/ 642         /                   /                                              /   109    \/ 642                                                            | |           /                                              /   109    \/ 642         /                   /                                              /   109    \/ 642                                                            |
|          /                   /                                        288*3 /   ----- + -------                                                                /                                        288*3 /   ----- + -------  | |          /                                        288*3 /   ----- + -------       /                   /                                        288*3 /   ----- + -------                                                           | |          /                                        288*3 /   ----- + -------       /                   /                                        288*3 /   ----- + -------                                                           | |          /                                        288*3 /   ----- + -------       /                   /                                        288*3 /   ----- + -------                                                           |
|    3   \/                  \/                                             \/    13824     576                                                                \/                                             \/    13824     576    | |    3   \/                                             \/    13824     576       \/                  \/                                             \/    13824     576                                                             | |    3   \/                                             \/    13824     576       \/                  \/                                             \/    13824     576                                                             | |    3   \/                                             \/    13824     576       \/                  \/                                             \/    13824     576                                                             |
|x + - - --------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------|*|x + - - ---------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------|*|x + - - ---------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------|*|x + - + ---------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------|
\    4                                                                            2                                                                                                              2                                   / \    4                                     2                                                                                                               2                                                                         / \    4                                     2                                                                                                               2                                                                         / \    4                                     2                                                                                                               2                                                                         /

$$\left(x + \left(- \frac{\sqrt{- \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{12} + 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}}{2} + \frac{3}{4} + \frac{\sqrt{- \frac{1}{\sqrt{- \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{12} + 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}} - 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}} + \frac{1}{6} + \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}}}{2}\right)\right) \left(x + \left(- \frac{\sqrt{- 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}} + \frac{1}{6} + \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{\sqrt{- \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{12} + 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}}}}{2} + \frac{\sqrt{- \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{12} + 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}}{2} + \frac{3}{4}\right)\right) \left(x + \left(- \frac{\sqrt{- \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{12} + 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}}{2} + \frac{3}{4} - \frac{\sqrt{- \frac{1}{\sqrt{- \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{12} + 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}} - 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}} + \frac{1}{6} + \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}}}{2}\right)\right) \left(x + \left(\frac{\sqrt{- \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{12} + 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}}{2} + \frac{3}{4} + \frac{\sqrt{- 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}} + \frac{1}{6} + \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{\sqrt{- \frac{71}{288 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}} + \frac{1}{12} + 2 \sqrt[3]{\frac{109}{13824} + \frac{\sqrt{642}}{576}}}}}}{2}\right)\right)$$