Definition: The zero product property states that if a product of factors equals zero, then at least one of the factors must equal zero.
Why is it true?
No two nonzero numbers multiply to give zero. So when a b = 0, either a is 0, or b is 0, or both are. This is the reason factoring solves equations.
How do you use it?
Move everything to one side so the other side is exactly 0. Factor. Set each factor equal to 0 and solve the small equations.
Example. Solve 2x2 + 5x = 12. Rewrite as 2x2 + 5x – 12 = 0, which factors as (2x – 3)(x + 4) = 0. Then 2x – 3 = 0 gives x = 3/2, and x + 4 = 0 gives x = -4. Both values satisfy the original equation.
What is the big warning?
The property only works with zero. If (x – 2)(x – 5) = 10, you cannot say x – 2 = 10 or x – 5 = 10. Expand, subtract 10, and factor again. Another error is dividing both sides by a variable factor, which loses a solution; factor instead.
Where to learn it
- Solving Quadratic Equations Four Ways – lesson
- Factoring Polynomials with Patterns and Grouping – lesson