The distributive property

The distributive property lets you rewrite a product over a sum:

`a · (x + b) = a · x + a · b`

It works because multiplication "distributes" over addition. The same number `a` multiplies each term in the bracket.

Expanding

When you have a number times a bracket, multiply the number by each term inside.

Example. `3 · (x + 4) = 3x + 12`. Example. `5 · (n + 2) = 5n + 10`.

Don't forget to multiply the second term too — that's the most common mistake.

Factoring (the reverse)

When two terms share the same factor, you can pull it out front:

`a · x + a · b = a · (x + b)`

Example. `3x + 12`. Both 3x and 12 are divisible by 3, so factor 3 out: `3 · (x + 4)`. Example. `5n + 10 = 5 · (n + 2)`.

How to spot the common factor

1. Look at the coefficient in front of the variable.
2. Look at the constant by itself.
3. Find the greatest common divisor of those two numbers — that's the factor you can pull out.

For `8x + 12` the GCD of 8 and 12 is 4, so `8x + 12 = 4 · (2x + 3)`.

Why this is useful

• It makes mental arithmetic easier: `7 · 23 = 7 · (20 + 3) = 140 + 21 = 161`.
• It turns ugly expressions into shorter ones.
• In later grades you'll use it for solving equations and simplifying algebraic fractions.