Lesson 4: Distribution and Factoring

Property

If $a, b, c$ are real numbers, then
$a(b+c) = ab+ac$
$(b+c)a = ba+ca$
$a(b-c) = ab-ac$
$(b-c)a = ba-ca$

Examples

• To simplify $7(x+2)$, distribute the $7$: $7 \cdot x + 7 \cdot 2 = 7x + 14$.
• To simplify $-5(y-4)$, distribute the $-5$: $(-5) \cdot y - (-5) \cdot 4 = -5y + 20$.
• To simplify $9 - 3(x+1)$, first distribute the $-3$: $9 - 3x - 3$. Then combine like terms to get $6 - 3x$.

Explanation

The distributive property lets you 'distribute' or 'pass out' the number outside the parentheses to every term inside. It's the key to removing parentheses and simplifying expressions in algebra. Think of it as sharing the multiplication.

Property

Distributive Property

If $a, b, c$ are real numbers, then

The form on the left is used to multiply. The form on the right is used to factor.

Property

1. Factor numerator and denominator completely.
2. Divide numerator and denominator by any common factors. We can cancel common factors (expressions that are multiplied together), but not common terms (expressions that are added or subtracted).

Examples