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

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Answer 1 · 2014-08-19T07:07:53

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

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

A problem could be as easy as a factored quadratic:

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

So:

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

or

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

Or it could be more complicated like:

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

So:

$sin x - \frac{1}{2} = 0$ $sin x-1/2=0$
$sin x = \frac{1}{2}$ $sin x = 1/2$
$x=pi/6+2pi n, n in ZZ$
$x=(5 pi)/6+2pi n, n in ZZ$

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.

If ab=0ab=0ab=0, what is true of aaa or bbb?