



If you
have two factors like:






(X + 1)(X
 1)






Look
what happens when you multiply them out ... 





(X + 1)(X
 1) = X^{2}  X + X 
1 = X^{2}  1






The
middle terms cancel each other out! 





What
we get at the end is called the difference of two squares. 


They
call it that because: 


It
is two terms. 


The
second one is subtracted. 


They
can both be gotten by multiplying something times itself. 





X × X = X^{2}
and 1 × 1 = 1 





When
we are factoring stuff, 


we
are going in the other direction. 





We
start with the multiplied out thing 


and
try to find the factors. 





Say
we have ... 


X^{2} 
9 





and
want to factor it. 





We
have no middle term. 


We
just have an X ^{2} and a number. 


The
number is negative. 


You
can get the number by multiplying 3 times itself. 





We
can factor this as: 


(X + 3)(X
 3)






Here's
a tougher one. 





Factor
this ... 


4X^{2} 
25 





We
have no middle term. 


The
number is negative. 


You
can get the number by multiplying 3 times itself. 


But
instead of X ^{2}, we have 4X
^{2}. 





That's
not a problem ... 





2X x 2X = 4X^{2}
and 5 x 5 = 25 





So
we can factor this ... 





4X^{2} 
25 = (2X + 5)(2X  5) 





In
general, the formula for doing these is: 











Here
are a few more examples ... 





4X^{2}
 9 = 
(2X + 3)(2X 
3) 


16X^{2}
 1 = 
(4X + 1)(4X 
1) 


X^{4}
 81 = 
(X^{2}
+9)(X^{2}
 9) = (X^{2}
+9)(X + 3)(X  3) 





We
got a little trickier on that last one! 


If
you have any big even power (like X
^{4}), 


you
can just chop the power in half in the factors (make them X
^{2}'s). 


If
one of those factors is still the difference of two squares, 


you
can chop it in half again. 





If
the numbers in the problem 


are
not squares of whole numbers, 


we
can still do this stuff. 





Here's
some examples of that ... 











As
long as we have a minus sign between the two terms, 


we
can use this trick to factor them. 





Could
we factor stuff like X ^{2} + 1? 





For
now, no. 


Later
on, you may learn about 


something
called IMAGINARY NUMBERS. 


They
can help to factor stuff like this. 


But
for now, the answer to: 


"Can
X ^{2} + 1 be factored?" is no. 





OK
here's another question. 


All
the powers we have talked about factoring so far are even. 


What
if we had an odd power? 





Could
we factor X ^{3}  1? 


Might
this be called the difference of two cubes???? 





Yup!
It is. 


And
on the next page, you will learn all about it. 





copyright 2005 Bruce Kirkpatrick 
