Algebra 1 Simplifying Polynomial Fractions
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Make It Simple
Simplifying Polynomial Fractions

 
 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

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