Zero Product Property

The zero product property says that if a product equals zero, at least one of its factors must be zero. It turns one hard equation into several easy ones. The catch is that it only works when one side is exactly 0 and the other side is written as factors multiplied together, so most of the work is getting the equation into that shape. Since in many cases, there can be more than 2 factors, we can generalize the zero product property as shown below.

Zero Product Property
The rule

If a · b = 0, then a = 0 or b = 0 (or both).

The general rule, for any number of factors

For any real numbers a, b, c, …, and Ω,

if a · b · c · … · Ω = 0,
then a = 0, b = 0, c = 0, …, or Ω = 0.

At least one of the factors must be 0. More than one can be 0 at the same time, and all of them can be 0. This rule is also called the zero-product principle.

Why it worksTwo numbers that aren't 0 never multiply to 0: 3 · 5 = 15 and −2 · 4 = −8. The only way to get a product of 0 is for at least one factor to be 0.
It only works when the product is 0If a · b = 6, you can't conclude that a = 6 or b = 6. The factors could be 2 and 3, or 1 and 6, or −2 and −3. Only a product of 0 tells you what one of the factors must be.
Four steps
  1. Get 0 alone on one side. Add or subtract terms until one side is 0.
  2. Factor the other side completely.
  3. Set each factor equal to 0 and solve. A number factor like 5 can never be 0, so it gives no solution.
  4. Check each answer in the original equation.

What must be zero?

Before solving any equations, get a feel for the rule itself. Each product below tells you something about its factors. Decide what it tells you.

Answered correctly: 0 of 5

Is it ready?

You can use the zero product property only when one side is 0 and the other side is a product. Decide for each equation: can you set the factors equal to 0 right now, or does it need work first?

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Worked examples

Each example is solved in full below. To work through one yourself first, press Step through it and reveal one step at a time.

Your turn

Solve this equationSolved: 0

Step 1

Is one side 0 and the other side a product of factors?

Does it work for matrices?

No. The zero product property is about real numbers. With matrices, two matrices that are not the zero matrix can multiply to give the zero matrix.

A =0036 B =0−201
A × B =0000

Neither A nor B is the zero matrix, yet A × B is. So a zero product of matrices doesn't tell you that one of the matrices is zero.

Each entry is a row of A times a column of B.
Row 1, column 1:0 · 0 + 0 · 0 = 0
Row 1, column 2:0 · (−2) + 0 · 1 = 0
Row 2, column 1:3 · 0 + 6 · 0 = 0
Row 2, column 2:3 · (−2) + 6 · 1 = −6 + 6 = 0
Every entry is 0, so A × B is the zero matrix.
What about B × A?Multiply the same two matrices in the opposite order.
B × A =−6−1236
Row 1:0 · 0 + (−2) · 3 = −6,   0 · 0 + (−2) · 6 = −12
Row 2:0 · 0 + 1 · 3 = 3,   0 · 0 + 1 · 6 = 6
Same two matrices, opposite order, and the product is not zero.Matrix multiplication depends on the order. That's another way matrices behave differently from numbers.

With real numbers, two numbers that aren't 0 never multiply to 0. Matrices don't have that guarantee, so the property can fail for them.

Common mistakes

  • Using it when the product isn't 0. x(x − 5) = 6 does not mean x = 6 or x − 5 = 6. Subtract 6 first, then factor.
  • Dividing both sides by x. In 4x² = 64x, dividing by x gives only x = 16 and loses x = 0. Move everything to one side and factor instead.
  • Saying all the factors equal 0. Only one factor has to be 0. In (x − 2)(x + 3) = 0, x is 2 or −3, never both at once.
  • Rushing the last step. Once you set each factor equal to 0, solve each small equation carefully. In (x + 5)(3x − 1) = 0, the solutions are x = −5 and x = 1/3, not 5 and 1.
  • Losing x = 0. In 2x(x − 7) = 0, the factor x gives the solution x = 0. Don't drop it.
  • Treating a number factor as a solution. In 2x(x − 7) = 0, the factor 2 can never equal 0, so it doesn't give a value of x. The solutions are only x = 0 and x = 7.
  • Trying it on a sum. (x − 2) + (x + 3) = 0 is not a product. Combine like terms instead.

How to use the zero product property to solve quadratic equations by factoring