



Now we
actually get to use this factoring for something! 





If we
have something like ... 











Can we
make this easier to deal with? 


That is,
can we make the fraction look simpler? 


As it
sits, no we can't. 


But if we
factor it ... 





Let's
see: 








We can
write this as: 








When we
have a fraction where all the things are multiplied 


like
factors, we can split up these factors into different fractions. 


Using a's
and b's, we might say ... 











When we
have stuff like this, 


the
factors don't care how they are split ... 











Let's see
what damage we can do to our problem 


with this
trick ... 











Now look
at those puppies we split apart. 


One has
the same term in the numerator and denominator. 


That
means it is equal to 1. 


So we can
say ... 











Multiplying
something times 1 doesn't change the value 


so we
don't even need to write it in if we don't want to. 


What we
do in real life when we have the same thing 


on the
top and the bottom of a fraction is to cancel. 











Remember
that to do this, 


everything
in the numerator and denominator 


must be
multiplied together. 





If we
have: 











We can
not do any canceling because of that 2. 





Let's see
another one: 





Example: 








Factoring
this we get ... 











Then
canceling ... 











OK. How
about this one ... 





Example: 











The
numerator is not a perfect square, 


but it
can be factored. 





The
denominator is the difference of 2 squares: 











Example: 





What
would we do with something like ... 











What we
do is this ... 











Now
before you start multiplying these things out 


try to
factor then all to see if there are any factors 


that you
can cancel. 





If you
work with it for a while, 


you will
find that all 4 of these terms will factor. 


Eventually
you get: 











Now since
everything is multiplied by everything else 


on the
top and on the bottom, 


we can
cancel any time we find the same term 


on the
top and on the bottom. 





Is there
anything to cancel? 


Yup. 


In fact,
everything cancels! 











Hey, if
EVERYTHING CANCELS, what's left? 





Remember
that before, when we started this canceling business, 


we said
that we were going to get rid of things 


that were
equal to one. 


What we
actually did here is: 











So as
weird as it looks ... 











Example: 





What
would you do with this one? 











There is
all kinds of fancy math hocus pocus to prove it, 


but the
deal is this ... 











So you
flip the thing you were dividing by 


and then
multiply ... 











Now let's
solve it! 











So we get
... 











Can we
factor this any farther? 





Sure. 


We could
factor the (X  3) as the difference of two squares. 


It would
be weird, but we know how. 


We could
factor the (X + 2) as the sum of two cubes. 


That's
even more weird. 


The
question is, why would we do it? 


We can't
cancel anything else 


so
there's no reason to do any more factoring. 





Should we
multiply it out? 





If you
have a reason to, sure. 


If not,
why bother. 





I can't
think of a reason to, can you? 





copyright 2005 Bruce Kirkpatrick 
