Lesson 4: Expanding Expressions using the Distributive Property

Property

If $a$, $b$, and $c$ are any numbers, then

If the terms inside parentheses are not like terms, we have no choice but to use the distributive law to simplify the expression.

Examples

• To simplify $5(x+4)$, distribute the 5 to each term: $5(x) + 5(4) = 5x + 20$.
• To simplify $-3(2y-1)$, multiply each inner term by $-3$: $-3(2y) - 3(-1) = -6y + 3$.
• To simplify $(a-5)6$, distribute the 6 from the right: $a(6) - 5(6) = 6a - 30$.

Explanation

The distributive law lets you multiply a number outside parentheses by each term inside. It's like sharing the outside number with every term in the group through multiplication. This is essential when you can't combine the terms inside first.

Property

When multiplying a number by a sum or difference in parentheses, you can distribute the multiplication to each term inside the parentheses.

For algebraic expressions:

Property

When you distribute a negative number, you must multiply the negative number by each term inside the parentheses.
Be careful to get the signs correct. Remember that $-a$ is equivalent to $-1 \cdot a$.

Examples

• To simplify $-2(4y + 1)$, distribute the $-2$: $(-2) \cdot 4y + (-2) \cdot 1$, which results in $-8y - 2$.

• To simplify $-11(4 - 3a)$, distribute the $-11$: $(-11) \cdot 4 - (-11) \cdot 3a$, which results in $-44 + 33a$.