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

Aug 19, 2014

$a = 0$ 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:

$\left(x - 3\right) \left(x + 2\right) = 0$

So:

$x - 3 = 0$
$x = 3$

or

$x + 2 = 0$
$x = - 2$

Or it could be more complicated like:

$\left(\sin x - \frac{1}{2}\right) \left(\cos x + \frac{1}{\sqrt{2}}\right) = 0$

So:

$\sin x - \frac{1}{2} = 0$
$\sin x = \frac{1}{2}$
$x = \frac{\pi}{6} + 2 \pi n , n \in \mathbb{Z}$
$x = \frac{5 \pi}{6} + 2 \pi n , n \in \mathbb{Z}$

or

$\cos x + \frac{1}{\sqrt{2}} = 0$
$\cos x = - \frac{1}{\sqrt{2}}$
$x = \frac{3 \pi}{4} + \pi n , n \in \mathbb{Z}$

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