distributive law. In mathematics, given any two operations, symbolized by * and ∘, the first operation, *, is distributive over the second, ∘, if a*(b∘c)=(a*b)∘(a*c) for all possible choices of a, b, and c. Multiplication, ×, is distributive over addition, +, since for any numbers a, b, and c, a×(b+c)=(a×b)+(a×c). For example, for the numbers 2, 3, and 4, 2×(3+4)=14 and (2×3)+(2×4)=14, meaning that 2×(3+4)=(2×3)+(2×4). Strictly speaking, this law expresses only left distributivity, i.e., a is distributed from the left side of (b+c); the corresponding definition for right distributivity is (a+b)×c=(a×c)+(b×c).

One of the laws relating to number operations. In symbols, it is stated: math.a(math.b + math.c) = math.amath.b + math.amath.c. The monomial factor math.a is distributed, or separately applied, to each term of the polynomial factor math.b + math.c, resulting in the product math.amath.b + math.amath.c. It can also be stated in words: The result of first adding several numbers and then multiplying the sum by some number is the same as first multiplying each separately by the number and then adding the products. Seealso associative law; commutative law.

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