In algebra, the zero-product property states that the product of two nonzero elements is nonzero.
This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties.
[1] All of the number systems studied in elementary mathematics — the integers
, the rational numbers
, the real numbers
, and the complex numbers
— satisfy the zero-product property.
In general, a ring which satisfies the zero-product property is called a domain.
is an algebraic structure.
have the zero-product property?
In order for this question to have meaning,
is a ring, though it could be something else, e.g. the set of nonnegative integers
with ordinary addition and multiplication, which is only a (commutative) semiring.
satisfies the zero-product property, and if
also satisfies the zero product property: if
can also be considered as elements of
are univariate polynomials with real coefficients, and
is a real number such that
to come from any integral domain.)
By the zero-product property, it follows that either
In other words, the roots of
Thus, one can use factorization to find the roots of a polynomial.
In general, suppose
is an integral domain and
is a monic univariate polynomial of degree
distinct roots
By the zero-product property, it follows that
distinct roots.
is not an integral domain, then the conclusion need not hold.
For example, the cubic polynomial