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 PropertyEdit
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 itEdit
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:
How to use itEdit
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!