A very important type of equation is one of the form
‘$\,xy = 0\,$’.
This type of equation has zero on one side (usually the right-hand side),
and things being multiplied on the other side.
How can a sentence of this form be true?
To answer this question, consider the following.
Suppose I were to say to you:
Can you tell me anything about the numbers I'm thinking of?
Indeed!
The only way that numbers can multiply to give zero is if
at least one of the numbers is zero:
That is, in order for the sentence
‘$\,xy = 0\,$’ to be true,
either
$\,x\,$ must equal $\,0\,$,
or $\,y\,$ must equal $\,0\,$,
or both must equal zero.
The ‘official’ statement of this property is called the Zero Factor Law.
Don't worry right now if the formal statement below doesn't make sense to you.
You'll learn how to “translate” it in upcoming sections.
With this idea in mind, consider the equation:
‘$\,x(x-1) = 0\,$’
The things being multiplied on the left-hand side are:
$\,x\,$ and $\,x-1\,$
In order for the equation to be true, either:
$\,x = 0\,$ or $\,x-1 = 0\,$
Consequently, the only numbers that make the equation true are
$\,0\,$ and $\,1\,$.
NOTE:
When presented with an equation like ‘$\,x(x-1) = 0\,$’ or
‘$\,(x-2)(x+3) = 0\,$’,
many people's first reaction is
to start multiplying out the left-hand side.
RESIST THIS TEMPTATION!
Multiplying out does absolutely no good (except perhaps giving
practice with multiplying polynomials).
The correct thought process is:
Ah hah!
I have things being MULTIPLIED
that equal ZERO.
So... one of the things being multiplied must equal zero!
In the web exercise, you must:
For more advanced students, a graph is displayed:
e.g., the equation $\,(x-1)(x-1)(x+2) = 0\,$
is optionally accompanied by the
graph of $\,y = (x-1)(x-1)(x+2)\,$.
Notice that you are finding the place(s) where this graph crosses the $x$-axis.
Click the “show/hide graph” button if you prefer not to see the graph.