Zero-product property

In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, it is the following assertion:
If formula_1, then formula_2 or formula_3.
The zero-product 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. All of the number systems studied in elementary mathematics — the integers formula_4, the rational numbers formula_5, the real numbers formula_6, and the complex numbers formula_7 — satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.
Algebraic context.
Suppose formula_8 is an algebraic structure. We might ask, does formula_8 have the zero-product property? In order for this question to have meaning, formula_8 must have both additive structure and multiplicative structure. Usually one assumes that formula_8 is a ring, though it could be something else, e.g. the set of nonnegative integers formula_12 with ordinary addition and multiplication, which is only a (commutative) semiring.
Note that if formula_8 satisfies the zero-product property, and if formula_14 is a subset of formula_8, then formula_14 also satisfies the zero product property: if formula_17 and formula_18 are elements of formula_14 such that formula_20, then either formula_21 or formula_22 because formula_17 and formula_18 can also be considered as elements of formula_8.
Application to finding roots of polynomials.
Suppose formula_51 and formula_52 are univariate polynomials with real coefficients, and formula_53 is a real number such that formula_54. (Actually, we may allow the coefficients and formula_53 to come from any integral domain.) By the zero-product property, it follows that either formula_56 or formula_57. In other words, the roots of formula_58 are precisely the roots of formula_51 together with the roots of formula_52.
Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial formula_61 factorizes as formula_62; hence, its roots are precisely 3, 1, and −2.
In general, suppose formula_63 is an integral domain and formula_64 is a monic univariate polynomial of degree formula_65 with coefficients in formula_63. Suppose also that formula_64 has formula_68 distinct roots formula_69. It follows (but we do not prove here) that formula_64 factorizes as formula_71. By the zero-product property, it follows that formula_72 are the "only" roots of formula_64: any root of formula_64 must be a root of formula_75 for some formula_76. In particular, formula_64 has at most formula_68 distinct roots.
If however formula_63 is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial formula_80 has six roots in formula_81 (though it has only three roots in formula_4).