Distributive Property
The distributive property states that a(b + c) = ab + ac, allowing multiplication to be distributed over addition or subtraction.
Formula
a(b + c) = ab + ac
Definition
The distributive property says that when you multiply a number by a group in parentheses, you multiply that number by each thing inside the parentheses separately: for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$, and it works the same way for subtraction, $a(b - c) = ab - ac$. It is used both to expand expressions and to factor them in reverse. In ring theory, multiplication distributing over addition, $a(b + c) = ab + ac$ and $(b + c)a = ba + ca$ (left and right distributivity), is one of the defining axioms of a ring; in a commutative ring both forms are identical, and the distributive law makes the ring a bimodule over itself.
Example
$3(x + 4) = 3 \times x + 3 \times 4 = 3x + 12$: the $3$ is distributed to both the $x$ and the $4$. Expand: $-2(3x - 5) = -6x + 10$; factor using the distributive property in reverse: $6x + 9 = 3(2x + 3)$. In the matrix ring $M_n(R)$, multiplication distributes over addition, $A(B + C) = AB + AC$, even though matrix multiplication is non-commutative, so $AB \neq BA$ in general.
Key Insight
Imagine handing out $3$ cookies to each person in a group of $(x + 4)$ people: everyone gets $3$ cookies, so you give out $3x + 12$ cookies total. The distributive property is the bridge between multiplication and addition in algebra, virtually all polynomial expansion relies on it and factoring is just applying it backward; distributivity is what connects the additive and multiplicative structures of a ring, and without it the two operations would be independent and algebra as we know it would not hold together.