nullprodukt

Nullprodukt is a term used in mathematics to denote a product that equals zero. In German texts it is often called Nullprodukt or Nullproduktregel. The concept is central to solving algebraic equations that arise from factoring expressions.

The zero-product property states that for real numbers and, more generally, in integral domains, if a·b = 0 then a = 0 or b = 0. Equivalently, a1·a2·...·an = 0 implies that at least one factor ai is zero. This property fails in rings with zero divisors, where nonzero elements can multiply to zero.

Examples include 0·x = 0 and 5·0 = 0. For a polynomial, (x+3)(x-3) = 0 has solutions x = -3 or x = 3. For a product of several factors, a·b·c = 0, the zero-product property holds in integral domains; in rings with zero divisors there exist nonzero a, b, c with abc = 0.

Applications of the nullprodukt concept include solving equations by factoring, finding polynomial roots, and justifying solution steps in algebra. The property underpins methods that set each factor equal to zero to identify potential solutions. In more general rings, one must be mindful of zero divisors, which can produce a zero product even when all factors are nonzero.

See also: zero divisor, integral domain, factorization, polynomial equation.