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Rule

Zero Product Property

If the product of two real numbers is zero, then one or both of the numbers is equal to zero.

If then or

This fact is also true if and are algebraic expressions.

Proof

Let and be two real numbers such that The goal is to show that at least one of them is zero. If or there is nothing to prove and the property is true. The interesting case is when one of the numbers is different from zero. Therefore, assume that is a non-zero number.
Since it has a multiplicative inverse, which is Next, multiply both sides of the left-hand equation by and simplify.
Simplify
It was obtained that Consequently, if the product is equal to and then This shows that when the product of two numbers is zero, at least one of them must be zero.