What is the Zero Product Property?
The Zero Product Property says that if the product of a multiplication problem is zero, then at least one of the factors has to be zero as well. The official definition is: “if \(ab=0\), then \(a=0\) or \(b=0\)”.
When you think about it, this property is kind of a no-brainer and it might seem unnecessary and redundant to have an entire property to describe how multiplication works with zeros.
However, it ends up being very useful when working with polynomials because it allows you to find the roots of polynomials, solve polynomials by factoring, and graph quadratics in factored form.
Examples
\[6z=0\]
The Zero Product Property says that at least one of the factors (\(\purple 6\) or \(\purple z\)) has to be zero because their product is zero.
Obviously, \(\purple 6\) is not zero, so that means that \(\purple z=0\) is the solution to this equation.
\[4xy=0\]
The Zero Product Property says that at least one of the factors (\(\blue 4\), \(\blue x\), or \(\blue y\)) has to be zero because their product is zero.
Obviously, \(\blue 4\) is not zero, so that means that either \(\blue x\) or \(\blue y\) has to be zero.
So, any ordered pair in the form of \(\blue (0, y)\) or \(\blue (x, 0)\) is a solution to this equation.
\[(x-4)(x+2)=0\]
The Zero Product Property says that at least one of the factors (\(\green x-4\) or \(\green x+2\)) has to be zero because their product is zero.
\(\green x-4=0\) if \(\green x=4\).
And \(\green x+2=0\) if \(\green x=-2\).
So, \(\green x=4\) and \(\green x=-2\) are the solutions to this equation.
When Will I Use the Zero Product Property?
When you are asked to find the roots of a polynomial, you will factor the polynomial and then set each of the factors equal to zero.
When you graph quadratics in factored form, you will use the Zero Product Property to find the x-intercepts of the graph.
