Question

If `ab=0`, what is true of `a` or `b`?

Answer 1

`a=0` or `b=0`. This does not exclude the case: `a=0` and `b=0`.

This property is used frequently to solve problems, even ones that are very complex.

A problem could be as easy as a factored quadratic:

`(x-3)(x+2)=0`

So:

`x-3=0`
`x=3`

or

`x+2=0`
`x=-2`

Or it could be more complicated like:

`(sin x-1/2)(cos x+1/sqrt(2))=0`

So:

`sin x-1/2=0`
`sin x = 1/2`
`x=pi/6+2pi n, n in ZZ`
`x=(5 pi)/6+2pi n, n in ZZ`

or

`cos x+1/sqrt(2)=0`
`cos x=-1/sqrt(2)`
`x=(3pi)/4+pi n, n in ZZ`

As you can see, the zero factor property allows us to algebraically solve many math problems.