# Distributive Factor Out

The `Distributive Property`

of numbers says that we can factor out common values from terms that are connected with an addition operator.

This rule is expressed by the equation `ab + ac = a(b + c)`

Note

This is a core transformation used in combining like terms, though we normally skip over it mentally because humans are quite smart.

Consider that the `9y + 9y`

example from above becomes `(9 + 9) * y`

. If you then apply a constant simplification rule, you end up with `18y`

, which is the result of combining the two like `y`

terms.

### Transformations¶

Given a common parent node, this rule extracts the common value from both sides, leaving an addition and a multiplication.

#### Addition¶

`ab + ac = a(b + c)`

```
+ *
/ \ / \
/ \ / \
/ \ -> / \
* * a +
/ \ / \ / \
a b a c b c
```

### Examples¶

Info

All the examples shown below are drawn from the mathy test suite that verifies the expected input/output combinations for rule transformations.

Input | Output | Valid |
---|---|---|

g + -x^3 + 4x^3 + 19p^4 + -1y | g + (-1 + 4) * x^3 + 19p^4 + -1y | ✔ |

5.8c + (3393c + 6o + -8614k) | (5.8 + 3393) * c + (6o + -8614k) | ✔ |

8c + (4v + 9n) + 7n + 4r | 8c + 4v + (9 + 7) * n + 4r | ✔ |

11 + 11d + (1d + d^2) | 11 + (11 + 1) * d + d^2 | ✔ |

5a + o + (o + y) + 4s | 5a + (1 + 1) * o + y + 4s | ✔ |

15 + 20j + 9j | 15 + (20 + 9) * j | ✔ |

23f + (5f + 5d) | (23 + 5) * f + 5d | ✔ |

9y + 9y | (9 + 9) * y | ✔ |

14x + 7x | (14 + 7) * x | ✔ |

6 + 4 | (3 + 2) * 2 | ✔ |

7 + 7 | (1 + 1) * 7 | ✔ |

4 + (z + 4) | --- | --- |

6 + 4 | --- | --- |

(z * 4 + z * 84x) + 1 | --- | --- |