Zero Product Property
The zero product property states that if a product of factors equals zero, then at least one of the factors must equal zero.
Formula
If ab = 0, then a = 0 or b = 0
Definition
The zero product property says that if two things multiplied together equal zero, then at least one of them must be zero: you cannot multiply two non-zero numbers and get zero. Formally, if $ab = 0$, then $a = 0$ or $b = 0$ (or both), a property essential for solving factored equations, though it only applies when the right side of the equation is truly zero. The property holds in an integral domain, a commutative ring $R$ with unity in which $ab = 0$ implies $a = 0$ or $b = 0$; elements $a$, $b$ with $ab = 0$ but neither equal to zero are called zero divisors. The integers $\mathbb{Z}$ and all fields are integral domains, but the ring $\mathbb{Z}/6\mathbb{Z}$ is not, since $2 \times 3 = 0 \bmod 6$ yet neither $2$ nor $3$ is zero.
Example
If $n \times 5 = 0$, then $n$ must be $0$. If $(x - 3)(x + 2) = 0$, then either $x - 3 = 0$ (so $x = 3$) or $x + 2 = 0$ (so $x = -2$); this is exactly how $x^2 - x - 6 = 0$ is solved, by factoring to $(x - 3)(x + 2) = 0$. In $\mathbb{Z}/6\mathbb{Z}$, $2 \times 3 = 6 = 0 \pmod{6}$, so $2$ and $3$ are zero divisors, meaning the zero product property fails there and factoring to solve equations is not valid in this ring without extra care.
Key Insight
Zero is the only number with this special "killer" property, multiplying anything by zero always produces zero, so always move all terms to one side and set the equation equal to zero before factoring to solve. Integral domains are precisely the commutative rings where the zero product property holds universally, which is what makes polynomial factoring a valid technique for finding roots.