Lesson 15: Using the Distributive Property to Simplify Expressions

New Concept

The Distributive Property lets you multiply a single value across terms inside parentheses, a key move for simplifying algebraic expressions.

The Distributive Property
For all real numbers $a, b, c$,

What’s next

Now, let's put it into practice. You'll work through examples with different number types and variables before applying it to a real-world problem.

Property

For all real numbers $a, b, c$, we have $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$.

Examples

$7(3 + 5) = 7(3) + 7(5) = 21 + 35 = 56$
$5(8 - 2) = 5(8) - 5(2) = 40 - 10 = 30$
$(6 - y)4 = 4(6) - 4(y) = 24 - 4y$

Explanation

Think of the number outside as a generous host! It 'distributes' itself by multiplying with every single term inside the party (the parentheses). Everyone gets a piece of the multiplication action, making sure no term is left out. This trick helps break down bigger problems into smaller, manageable parts.

Property

A negative sign outside parentheses, like $-(a+b)$, is the same as multiplying by $-1$. So, $-(a+b) = -1(a+b) = -a - b$.

Examples

$-(8 + 3) = (-1)(8) + (-1)(3) = -8 - 3 = -11$
$-7(-5 - 2) = (-7)(-5) + (-7)(-2) = 35 + 14 = 49$
$-5(x + 2) = (-5)(x) + (-5)(2) = -5x - 10$

Explanation

When a negative number crashes the party, it flips the sign of everyone inside the parentheses! It's a sneaky little trick where every positive becomes a negative and every negative becomes a positive after the multiplication. It’s a total sign-switching bonanza inside the parentheses, so pay close attention!

Property

Use the Distributive Property with variables and remember to add exponents when multiplying powers with the same base: $x^a \cdot x^b = x^{a+b}$.

Examples

$(9 - y)8 = 8(9) + 8(-y) = 72 - 8y$
$ab(ac + ad) = (ab)(ac) + (ab)(ad) = a^2bc + a^2bd$
$-yz(y^3 - yz) = (-yz)(y^3) + (-yz)(-yz) = -y^4z + y^2z^2$

Explanation

Distributing works just the same with variables and letters. Multiply the term on the outside with each term on the inside. If you multiply variables with the same base, their exponents team up by adding their powers together! This makes simplifying long algebraic expressions a piece of cake.