Общий множитель 12*x^3*y+5*x^2*y-3*x*y^2+2*x*y-y^3

Комбинаторика

  /   2            2       3        \
y*\- y  + 2*x + 5*x  + 12*x  - 3*x*y/

$$y \left(12 x^{3} + 5 x^{2} - 3 x y + 2 x - y^{2}\right)$$

Объединение рациональных выражений

  /   2                             \
y*\- y  + x*(2 - 3*y + x*(5 + 12*x))/

$$y \left(x \left(x \left(12 x + 5\right) - 3 y + 2\right) - y^{2}\right)$$

Разложение на множители

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|                /                /                      2                                                                                                                           | |                /                /                      2                                                                                                                                                           | |                /                /                      2                                                                                                                                                           |  
|               /                /  /           2       \                   3                                                                                                        | |               /                /  /           2       \                   3                                                                                                                                        | |               /                /  /           2       \                   3                                                                                                                                        |  
|              /                /   |  415   9*y    15*y|      /   47   3*y\                                                                                                         | |              /                /   |  415   9*y    15*y|      /   47   3*y\                                                                                                                                         | |              /                /   |  415   9*y    15*y|      /   47   3*y\                                                                                                                                         |  
|             /                /    |- --- - ---- + ----|  - 4*|- --- + ---|        2                                                                                                | |             /                /    |- --- - ---- + ----|  - 4*|- --- + ---|        2         /          ___\                                                                                                        | |             /                /    |- --- - ---- + ----|  - 4*|- --- + ---|        2         /          ___\                                                                                                        |  
|            /        415    \/     \  864    4      16 /      \  144    4 /     9*y    15*y                                            47   3*y                                     | |            /        415    \/     \  864    4      16 /      \  144    4 /     9*y    15*y  |  1   I*\/ 3 |                                                   47   3*y                                             | |            /        415    \/     \  864    4      16 /      \  144    4 /     9*y    15*y  |  1   I*\/ 3 |                                                   47   3*y                                             |  
|         3 /       - ---- + ------------------------------------------------- - ---- + ----                                         - --- + ---                                     | |         3 /       - ---- + ------------------------------------------------- - ---- + ---- *|- - - -------|                                                - --- + ---                                             | |         3 /       - ---- + ------------------------------------------------- - ---- + ---- *|- - + -------|                                                - --- + ---                                             |  
|    5    \/          1728                           2                            8      32                                            144    4                                      | |    5    \/          1728                           2                            8      32   \  2      2   /                                                  144    4                                              | |    5    \/          1728                           2                            8      32   \  2      2   /                                                  144    4                                              |  
|x + -- + ----------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------|*|x + -- + --------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------|*|x + -- + --------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------|*y
|    36                                            3                                                       __________________________________________________________________________| |    36                                                    3                                                                               __________________________________________________________________________| |    36                                                    3                                                                               __________________________________________________________________________|  
|                                                                                                         /                ___________________________________________               | |                                                                                                                                         /                ___________________________________________               | |                                                                                                                                         /                ___________________________________________               |  
|                                                                                                        /                /                      2                                   | |                                                                                                                                        /                /                      2                                   | |                                                                                                                                        /                /                      2                                   |  
|                                                                                                       /                /  /           2       \                   3                | |                                                                                                                                       /                /  /           2       \                   3                | |                                                                                                                                       /                /  /           2       \                   3                |  
|                                                                                                      /                /   |  415   9*y    15*y|      /   47   3*y\                 | |                                                                                                                                      /                /   |  415   9*y    15*y|      /   47   3*y\                 | |                                                                                                                                      /                /   |  415   9*y    15*y|      /   47   3*y\                 |  
|                                                                                                     /                /    |- --- - ---- + ----|  - 4*|- --- + ---|        2        | |                                                                                                                 /          ___\     /                /    |- --- - ---- + ----|  - 4*|- --- + ---|        2        | |                                                                                                                 /          ___\     /                /    |- --- - ---- + ----|  - 4*|- --- + ---|        2        |  
|                                                                                                    /        415    \/     \  864    4      16 /      \  144    4 /     9*y    15*y | |                                                                                                                 |  1   I*\/ 3 |    /        415    \/     \  864    4      16 /      \  144    4 /     9*y    15*y | |                                                                                                                 |  1   I*\/ 3 |    /        415    \/     \  864    4      16 /      \  144    4 /     9*y    15*y |  
|                                                                                               3*3 /       - ---- + ------------------------------------------------- - ---- + ---- | |                                                                                                               3*|- - - -------|*3 /       - ---- + ------------------------------------------------- - ---- + ---- | |                                                                                                               3*|- - + -------|*3 /       - ---- + ------------------------------------------------- - ---- + ---- |  
\                                                                                                 \/          1728                           2                            8      32  / \                                                                                                                 \  2      2   / \/          1728                           2                            8      32  / \                                                                                                                 \  2      2   / \/          1728                           2                            8      32  /  

$$y \left(x + \left(\frac{\frac{3 y}{4} - \frac{47}{144}}{3 \sqrt[3]{- \frac{9 y^{2}}{8} + \frac{15 y}{32} + \frac{\sqrt{- 4 \left(\frac{3 y}{4} - \frac{47}{144}\right)^{3} + \left(- \frac{9 y^{2}}{4} + \frac{15 y}{16} - \frac{415}{864}\right)^{2}}}{2} - \frac{415}{1728}}} + \frac{\sqrt[3]{- \frac{9 y^{2}}{8} + \frac{15 y}{32} + \frac{\sqrt{- 4 \left(\frac{3 y}{4} - \frac{47}{144}\right)^{3} + \left(- \frac{9 y^{2}}{4} + \frac{15 y}{16} - \frac{415}{864}\right)^{2}}}{2} - \frac{415}{1728}}}{3} + \frac{5}{36}\right)\right) \left(x + \left(\frac{\frac{3 y}{4} - \frac{47}{144}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{- \frac{9 y^{2}}{8} + \frac{15 y}{32} + \frac{\sqrt{- 4 \left(\frac{3 y}{4} - \frac{47}{144}\right)^{3} + \left(- \frac{9 y^{2}}{4} + \frac{15 y}{16} - \frac{415}{864}\right)^{2}}}{2} - \frac{415}{1728}}} + \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{- \frac{9 y^{2}}{8} + \frac{15 y}{32} + \frac{\sqrt{- 4 \left(\frac{3 y}{4} - \frac{47}{144}\right)^{3} + \left(- \frac{9 y^{2}}{4} + \frac{15 y}{16} - \frac{415}{864}\right)^{2}}}{2} - \frac{415}{1728}}}{3} + \frac{5}{36}\right)\right) \left(x + \left(\frac{\frac{3 y}{4} - \frac{47}{144}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{- \frac{9 y^{2}}{8} + \frac{15 y}{32} + \frac{\sqrt{- 4 \left(\frac{3 y}{4} - \frac{47}{144}\right)^{3} + \left(- \frac{9 y^{2}}{4} + \frac{15 y}{16} - \frac{415}{864}\right)^{2}}}{2} - \frac{415}{1728}}} + \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{- \frac{9 y^{2}}{8} + \frac{15 y}{32} + \frac{\sqrt{- 4 \left(\frac{3 y}{4} - \frac{47}{144}\right)^{3} + \left(- \frac{9 y^{2}}{4} + \frac{15 y}{16} - \frac{415}{864}\right)^{2}}}{2} - \frac{415}{1728}}}{3} + \frac{5}{36}\right)\right)$$

Собрать выражение

   3        2                2         3
- y  - 3*x*y  + 2*x*y + 5*y*x  + 12*y*x 

$$12 x^{3} y + 5 x^{2} y - 3 x y^{2} + 2 x y - y^{3}$$

   3     /         2       3\        2
- y  + y*\2*x + 5*x  + 12*x / - 3*x*y 

$$- 3 x y^{2} - y^{3} + y \left(12 x^{3} + 5 x^{2} + 2 x\right)$$

   3     /     2      \        2         3
- y  + x*\- 3*y  + 2*y/ + 5*y*x  + 12*y*x 

$$12 x^{3} y + 5 x^{2} y + x \left(- 3 y^{2} + 2 y\right) - y^{3}$$

Общее упрощение

  /   2            2       3        \
y*\- y  + 2*x + 5*x  + 12*x  - 3*x*y/

$$y \left(12 x^{3} + 5 x^{2} - 3 x y + 2 x - y^{2}\right)$$