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: 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_1, the rational numbers formula_2, the real numbers formula_3, and the complex numbers formula_4 — satisfy the zero-product property. In general, a ring which satisfies the  zero-product property is called a domain. Algebraic context. Suppose formula_5 is an algebraic structure. We might ask, does formula_5 have the zero-product property? In order for this question to have meaning, formula_5 must have both additive structure and multiplicative structure. Usually one assumes that formula_5 is a ring, though it could be something else, e.g. the set of nonnegative integers formula_9 with ordinary addition and multiplication, which is only a (commutative) semiring. Note that if formula_5 satisfies the zero-product property, and if formula_11 is a subset of formula_5, then formula_11 also satisfies the zero product property: if  formula_14 and formula_15 are elements of formula_11 such that formula_17, then either formula_18 or formula_19 because formula_14 and formula_15 can also be considered as elements of formula_5. Application to finding roots of polynomials. Suppose formula_48 and formula_49 are univariate polynomials with real coefficients, and formula_50 is a real number such that formula_51. (Actually, we may allow the coefficients and formula_50 to come from any integral domain.) By the zero-product property, it follows that either formula_53 or formula_54. In other words, the roots of formula_55 are precisely the roots of formula_48 together with the roots of formula_49. Thus, one can  use factorization to find the roots of a polynomial. For example, the polynomial formula_58 factorizes as formula_59; hence, its roots are precisely 3, 1, and −2. In general, suppose formula_60 is an integral domain and formula_61 is a monic univariate polynomial of degree formula_62 with coefficients in formula_60. Suppose also that formula_61 has formula_65 distinct roots formula_66. It follows (but we do not prove here) that formula_61 factorizes as formula_68. By the zero-product property, it follows that formula_69 are the "only" roots of formula_61: any root of formula_61 must be a root of formula_72 for some formula_73. In particular , formula_61 has at most formula_65 distinct roots. If however formula_60 is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial formula_77 has six roots in formula_28 (though it has only three roots in formula_1).