Expanding and Factoring

Student Summary

Properties of operations can be used in different ways to rewrite expressions and create equivalent expressions. For example, the distributive property can be used to expand an expression such as 3(x+5)3(x+5) to get 3x+153x+15.

The distributive property can also be used in the other direction to factor an expression such as 12x−812x−8. In this case, we know the product and need to find the factors.

The terms of the product go inside:

Area diagram, 1 row, 2 columns. Inside the diagram, starting with the first box, 12 x, negative 8. To the left of the row and above each column, blank boxes indicate missing labels.

Think of a factor each term has in common: 12x12x and −8-8 each have a factor of 4. The common factor can be placed on one side of the large rectangle:

Area diagram, 1 row, 2 columns.  To the left of the row, 4. Inside the boxes, 12 x, negative 8. Above each row a blank box indicates a missing label.

Now think: "4 times what is 12x?" and "4 times what is -8?" Write the other factors on the other side of the rectangle:

Area diagram, 1 row, 2 columns. To the left of the row, 4. Above each column, 3 x, negative 2. Beginning with the first box, 12 x, negative 8.

So, 12x−812x-8 is equivalent to 4(3x−2)4(3x-2).

Visual / Anchor Chart