Zero Product Property

The zero product property is an extremely important property that helps split up complicated equations into simpler ones. In order to apply the zero product property, one side of your equation must be a product of factors and the other side of the equation must be zero.

Zero Product Property

Consider the $n$ numbers, functions, or expressions $a_1,a_2,\dots ,a_n.$ If

$$a_1{a}_2{a}_3\cdots a_n=0,$$

then $a_1=0\text{ or }{a}_2=0\text{ or }{a}_3=0\text{ or }\dots \text{ or }{a}_n=0.$

Let’s look at an example of how this can be used to solve a complicated equation by breaking it down into simpler equations.

Example

Consider the following equation.

$$(\sqrt{x}-4)(x+3)(|x|-4)=0$$

According to the zero product property, we can split this up into three simpler equations.

$$\sqrt{x}-4=0\text{ or }x+3=0\text{ or }|x|-4=0$$

The solutions to each of these three equations is fairly easy to determine by inspection.

• The solution to to the equation $\sqrt{x}-4=0$ is $x=16$.
• The solution to the equation $x+3=0$ is $x=-3$.
• The solutions to the equation $|x|-4=0$ are $x=-4$ and $x=4.$

Therefore, the original equation has four solutions in total, which I am choosing to list below in ascending order.

$$\boxed{x=\left\{-4,-3,4,16\right\}}$$