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:

  • (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 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!

Proving the Zero Products PropertyEdit

Practice ProblemsEdit

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