One of the most useful tricks in all of math: multiplying a sum is the same as multiplying each part separately, then adding. Split a rectangle to see exactly why.
Picture a rectangle with height a and width (b + c). You can find its area two ways: multiply the whole width at once, or split it into two smaller rectangles and add their areas. Either way, you get the same total.
The same idea works with subtraction — multiply each part, then subtract instead of add.
For example: 5 × (12 − 4) the direct way is 5 × 8 = 40. Distributed, it's (5 × 12) − (5 × 4) = 60 − 20 = 40. Same answer either way.
This is where the distributive property becomes one of algebra's most-used tools. When a parenthesis has a variable inside, distribute the multiplier to every term.
For example: 3(x + 4) distributes to 3x + 12 — multiply the 3 by the x, and the 3 by the 4.
Distributing is the first step whenever you need to combine like terms in an expression with parentheses — you'll use this constantly in the next lesson.
Eight questions mixing numeric and algebraic distribution.