distributive property
Imagine buying 3 gift bags, each holding an apple and a banana. You can count by bag — 3 bags of (apple + banana) — or by fruit — 3 apples plus 3 bananas. Either way you have the same fruit. The distributive property captures this: a(b + c) = ab + ac. Multiplication spreads, or distributes, across an addition.
This single rule links multiplication and addition, and it is the bridge between arithmetic and algebra. It lets you expand 4(x + 2) into 4x + 8, and run in reverse it lets you factor 6x + 9 into 3(2x + 3). Mental math leans on it too: 7 × 103 = 7 × (100 + 3) = 700 + 21 = 721.
It works the same with subtraction, a(b − c) = ab − ac, and you must distribute to every term inside the parentheses — forgetting one is a classic slip, as in writing 4(x + 2) = 4x + 2. Note that distribution is one-directional: multiplication distributes over addition, but addition does not distribute over multiplication.
Expand 5(2x − 3): distribute the 5 to each term, 5 × 2x = 10x and 5 × (−3) = −15, giving 10x − 15.
Multiply the outside factor by every term inside the parentheses, keeping signs.
Distributing a negative flips every sign inside: −(x − 5) = −1(x − 5) = −x + 5. Treat the minus sign as a hidden factor of −1 and distribute it to each term.