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 a b = 0 {displaystyle ab=0} , then a = 0 {displaystyle a=0} or b = 0 {displaystyle b=0} . 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 or the nonexistence of nontrivial zero divisors. All of the number systems studied in elementary mathematics — the integers Z {displaystyle mathbb {Z} } , the rational numbers Q {displaystyle mathbb {Q} } , the real numbers R {displaystyle mathbb {R} } , and the complex numbers C {displaystyle mathbb {C} } — satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain. Suppose A {displaystyle A} is an algebraic structure. We might ask, does A {displaystyle A} have the zero-product property? In order for this question to have meaning, A {displaystyle A} must have both additive structure and multiplicative structure. Usually one assumes that A {displaystyle A} is a ring, though it could be something else, e.g. the set of nonnegative integers { 0 , 1 , 2 , … } {displaystyle {0,1,2,ldots }} with ordinary addition and multiplication, which is only a (commutative) semiring. Note that if A {displaystyle A} satisfies the zero-product property, and if B {displaystyle B} is a subset of A {displaystyle A} , then B {displaystyle B} also satisfies the zero product property: if a {displaystyle a} and b {displaystyle b} are elements of B {displaystyle B} such that a b = 0 {displaystyle ab=0} , then either a = 0 {displaystyle a=0} or b = 0 {displaystyle b=0} because a {displaystyle a} and b {displaystyle b} can also be considered as elements of A {displaystyle A} . Suppose P {displaystyle P} and Q {displaystyle Q} are univariate polynomials with real coefficients, and x {displaystyle x} is a real number such that P ( x ) Q ( x ) = 0 {displaystyle P(x)Q(x)=0} . (Actually, we may allow the coefficients and x {displaystyle x} to come from any integral domain.) By the zero-product property, it follows that either P ( x ) = 0 {displaystyle P(x)=0} or Q ( x ) = 0 {displaystyle Q(x)=0} . In other words, the roots of P Q {displaystyle PQ} are precisely the roots of P {displaystyle P} together with the roots of Q {displaystyle Q} . Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial x 3 − 2 x 2 − 5 x + 6 {displaystyle x^{3}-2x^{2}-5x+6} factorizes as ( x − 3 ) ( x − 1 ) ( x + 2 ) {displaystyle (x-3)(x-1)(x+2)} ; hence, its roots are precisely 3, 1, and -2. In general, suppose R {displaystyle R} is an integral domain and f {displaystyle f} is a monic univariate polynomial of degree d ≥ 1 {displaystyle dgeq 1} with coefficients in R {displaystyle R} . Suppose also that f {displaystyle f} has d {displaystyle d} distinct roots r 1 , … , r d ∈ R {displaystyle r_{1},ldots ,r_{d}in R} . It follows (but we do not prove here) that f {displaystyle f} factorizes as f ( x ) = ( x − r 1 ) ⋯ ( x − r d ) {displaystyle f(x)=(x-r_{1})cdots (x-r_{d})} . By the zero-product property, it follows that r 1 , … , r d {displaystyle r_{1},ldots ,r_{d}} are the only roots of f {displaystyle f} : any root of f {displaystyle f} must be a root of ( x − r i ) {displaystyle (x-r_{i})} for some i {displaystyle i} . In particular, f {displaystyle f} has at most d {displaystyle d} distinct roots. If however R {displaystyle R} is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial x 3 + 3 x 2 + 2 x {displaystyle x^{3}+3x^{2}+2x} has six roots in Z 6 {displaystyle mathbb {Z} _{6}} (though it has only three roots in Z {displaystyle mathbb {Z} } ).