Zero Products Property

The Zero Products Property is one of the most fundamentally important properties in Algebra. It is used all the time in solving quadratic equation.

Zero Products Property
If (x)(y)=0, then x=0 or y=0

This is a rather simple idea, but difficult to prove. One approach to proving it can be seen below. But first: When and how to use it. Here goes:

When to use it
You use the Zero Products Property when - well - when you have products that equal zero. The products are not limited to monomials, or just two products. For example you can have:


 * (x+2)(x-4)


 * (3x^2-6x+3)(6x-2)(x)


 * (x-2)(x-3)(x-4)(x-5)(x+7)(x^10+4x^6+9x+10+14x^-1)

How to use it
The Zero Products Property says if xy=0, then x=0 or y=0. So if (x+2)(x-3)=0, then (x+2)=0 or (x-3)=0. So in order to use it, you have to set all the products of the equation equal to zero. Here's a quick demenstration:

Solve this equation: 4(4x+3)(x-2)(x^2+4)x=0

So we need to set all the solutions to zero. Here we go:


 * 4=0. This equation does not have a solution.


 * 4x+3=0; 4x=-3, x=-3/4. The solution for this equation is -3/4


 * x-2=0; x=2. The soulution for this equation is x=2


 * x^2+4=0; x^2=-4. This equation does not have a real solution, because you can't have negative numbers come out of a square.


 * x=0 The solution for this equation is 0. Don't forget to do this, the single x looks unimportant and may be overlooked, but it still is another solution!

So the combined solution is: x=-3/4,0,2. That's the Zero Products Property!