Algebra 1 Difference of Two Squares
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What's the Difference
Difference of Two Squares

 

 If you have two factors like:

 

 (X + 1)(X - 1)

 
 Look what happens when you multiply them out ...
 

 (X + 1)(X - 1)    =    X2 - X + X - 1  =    X2 - 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 = X2    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 ...

X2 - 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 ...

4X2 - 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 = 4X2    and   5 x 5 = 25

 
 So we can factor this ...
 

4X2 - 25 = (2X + 5)(2X - 5)

 
 In general, the formula for doing these is:
 

 
 Here are a few more examples ...
 
4X2 - 9 =   (2X + 3)(2X - 3)
16X2 - 1 =   (4X + 1)(4X - 1)
 X4 - 81 =   (X2 +9)(X2 - 9) = (X2 +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

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