Zero Product Property
In short: The zero product property says that when a product of factors comes out to zero, some factor in that product has to be zero itself.
Algebra 1AlgebraA-SSE.B.3a
What is Zero Product Property?
Two nonzero numbers can never multiply to zero, and that fact is the zero product property: when \(ab = 0\), either \(a = 0\) or \(b = 0\), or both.
It turns factored equations into easy ones. In \((x – 4)(x + 7) = 0\), set each factor to zero separately to find \(x = 4\) or \(x = -7\). Zero on the right is essential; any other number there leaves the rule powerless.
How to spot it
- An equation shows factors in parentheses multiplied together and equal to 0, the cue for the zero product property.
- A quadratic has been set to zero and can be split into two binomials.
- The directions ask for all solutions, or the zeros, of a factored expression.
- One side of an equation is zero, and the other is a product such as \(x(x – 7)\).
The method
- Gather all terms on the left until the right side reads exactly 0.
- Factor the expression completely.
- Apply the zero product property by turning each factor into its own mini-equation equal to zero.
- Substitute every answer back, and drop any that clash with the situation.
Zero Product Property: solved example
Problem
A photo print runs 3 inches longer than its width, and the paper covers 54 square inches. How wide is it?
Solution
With width w and length \(w + 3\), the area gives \(w(w + 3) = 54\), so \(w^2 + 3w – 54 = 0\). Factor: \((w + 9)(w – 6) = 0\).
By the property, \(w = -9\) or \(w = 6\). Lengths are never negative, so the print measures 6 inches across by 9 inches tall, and \(6 \times 9 = 54\).
Try it
Question 1
Solve \((2x – 1)(x + 8) = 0\).
Show the answer
\(2x – 1 = 0\) gives \(x = \frac{1}{2}\); \(x + 8 = 0\) gives \(x = -8\).
Question 2
Solve \(x^2 – 7x = 0\).
Show the answer
Factor \(x(x – 7) = 0\), so \(x = 0\) or \(x = 7\).
Zero Product Property vs. Factoring
Factoring is the technique that rewrites a sum as a product. The property is the next step: it uses that product to read off solutions. A common error is to apply it to \((x – 2)(x + 3) = 10\) by setting each factor to 10; the right side must be zero first.
Keep going
Lessons
Worksheets and quizzes
- Solving Quadratic Equations – Worksheet
- Solving Quadratic Equations – Quiz
- Factoring Polynomials – Worksheet
- Factoring Polynomials – Quiz
Related words
- Factoring
- Zeros of a Function
- Quadratic Formula
- Factoring Out the GCF
- Difference of Squares
- X-Intercept
Whole dictionary: A–Z · Algebra 1 · Algebra