# Lesson 19

Expanding and Factoring

Let's use the distributive property to write expressions in different ways.

### 19.1: Number Talk: Parentheses

Find the value of each expression mentally.

\(2 + 3 \boldcdot 4\)

\((2+3)(4)\)

\(2 - 3 \boldcdot 4\)

\(2 - (3 + 4)\)

### 19.2: Factoring and Expanding with Negative Numbers

In each row, write the equivalent expression. If you get stuck, use a diagram to organize your work. The first row is provided as an example. Diagrams are provided for the first three rows.

factored | expanded |
---|---|

\(\text- 3(5 - 2y)\) | \(\text- 15 + 6y\) |

\(5(a-6)\) | |

\(6a-2b\) | |

\(\text- 4(2w-5z)\) | |

\(\text- (2x-3y)\) | |

\(20x-10y+15z\) | |

\(k(4-17)\) | |

\(10a-13a\) | |

\(\text- 2x(3y-z)\) | |

\(ab-bc-3bd\) | |

\(\text- x(3y-z+4w)\) |

Expand to create an equivalent expression that uses the fewest number of terms: \(\left(\left(\left(\left(x\strut+1\right)\frac12\right)+1\right)\frac12\right)+1\). If we wrote a new expression following the same pattern so that there were 20 sets of parentheses, how could it be expanded into an equivalent expression that uses the fewest number of terms?

### Summary

We can use properties of operations in different ways to rewrite expressions and create equivalent expressions. We have already seen that we can use the distributive property to **expand** an expression, for example \(3(x+5) = 3x+15\). We can also use the distributive property in the other direction and **factor** an expression, for example \(8x+12 = 4(2x+3)\).

We can organize the work of using distributive property to rewrite the expression \(12x-8\). In this case we know the product and need to find the factors.

The terms of the product go inside:

We look at the expressions and think about a factor they have in common. \(12x\) and \(\text- 8\) each have a factor of 4. We place the common factor on one side of the large rectangle:

Now we think: "4 times *what* is 12\(x\)?" "4 times *what* is -8?" and write the other factors on the other side of the rectangle:

So, \(12x-8\) is equivalent to \(4(3x-2)\).

### Glossary Entries

**expand**To expand an expression, we use the distributive property to rewrite a product as a sum. The new expression is equivalent to the original expression.

For example, we can expand the expression \(5(4x+7)\) to get the equivalent expression \(20x + 35\).

**factor (an expression)**To factor an expression, we use the distributive property to rewrite a sum as a product. The new expression is equivalent to the original expression.

For example, we can factor the expression \(20x + 35\) to get the equivalent expression \(5(4x+7)\).

**term**A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression \(5x + 18\) has two terms. The first term is \(5x\) and the second term is 18.