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Zero Product Property

The Zero Product Property states that if two numbers multiplied together equal zero, then at least one of those numbers must be zero. In algebra, this property is the reason you can solve equations by factoring — once you write an equation as a product equal to zero, you set each factor equal to zero and solve.

The Zero Product Property is an algebraic principle that applies to real numbers: if ab=0ab = 0, then a=0a = 0 or b=0b = 0 (or both). This property holds because zero is the only real number that, when multiplied by any other number, produces zero. It extends to products of more than two factors: if abc=0abc = 0, then at least one of aa, bb, or cc must equal zero. The property is fundamental to solving polynomial equations by factoring.

Key Formula

If ab=0, then a=0 or b=0\text{If } ab = 0, \text{ then } a = 0 \text{ or } b = 0
Where:
  • aa = any real number or expression
  • bb = any real number or expression

Worked Example

Problem: Solve the equation x² + 2x − 15 = 0 using the Zero Product Property.
Step 1: Factor the left side of the equation. Find two numbers that multiply to −15 and add to 2. Those numbers are 5 and −3.
(x+5)(x−3)=0(x + 5)(x - 3) = 0
Step 2: Apply the Zero Product Property. Since the product of the two factors equals zero, at least one factor must be zero.
x+5=0orx−3=0x + 5 = 0 \quad \text{or} \quad x - 3 = 0
Step 3: Solve each equation separately.
x=−5orx=3x = -5 \quad \text{or} \quad x = 3
Step 4: Check by substituting back. For x = −5: (−5)² + 2(−5) − 15 = 25 − 10 − 15 = 0. For x = 3: (3)² + 2(3) − 15 = 9 + 6 − 15 = 0. Both solutions work.
Answer: The solutions are x=−5x = -5 and x=3x = 3.

Why It Matters

The Zero Product Property is the key reason factoring works as a method for solving equations. Without it, writing an equation in factored form wouldn't actually help you find solutions. You'll rely on this property throughout algebra whenever you solve quadratic equations, higher-degree polynomials, or any equation where one side can be factored and the other side is zero.

Common Mistakes

Mistake: Applying the property when the product does not equal zero. For example, trying to solve (x + 2)(x − 1) = 6 by setting x + 2 = 6 and x − 1 = 6.
Correction: The Zero Product Property only works when the product equals zero. If the right side isn't zero, you need to expand, rearrange so one side is zero, and then factor again.
Mistake: Forgetting to set each factor equal to zero and only solving one of them.
Correction: Every factor could potentially be zero, so you must set each factor equal to zero and solve. Missing a factor means missing a solution.

Related Terms