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- Let's add some more rational expressions.
- So let's say I have 1 over x squared minus 9.
- And to that I'm going to add 2 over x squared
- plus 5x, plus 6.
- So just like we did when we first learned to add
- fractions, or to add rational numbers, we always had to come
- up with a common denominator.
- So once again, we're going to have to come up with a common
- denominator.
- And what we really want to do is find the least common
- multiple of these two expressions.
- And when I say that, let me just remind you.
- Let's say we wanted to add 1/4 plus 1/6.
- Well, we know that 1/4 is 1 over 2 times 2, and that 1/6
- is 1 over 2 times 3.
- So to be able to add these two things, we incrementally-- we
- both have a common 2.
- We both have this common 2.
- But in order for this guy to go into whatever our common
- denominator is, we're going to have to be
- divisible by another 2.
- So let me do that.
- So our common denominator's going to have to be-- we have
- this 2 in each of them.
- So that takes care of that first 2.
- But then we need the second 2.
- We need the second 2 there.
- 2 times a second 2.
- And then we need this 3 here.
- We need this 3 here.
- We have to have all of the prime factors of both of them.
- This has two 2's, so this guy has to have two 2's.
- This guy has a 2 and a 3, so this guy has to
- have a 2 and a 3.
- And so our least common multiple here is 2 times 2,
- times 3, which is 12.
- And then we can add accordingly.
- That is our common denominator.
- That's the least common multiple.
- You could have just multiplied these two out, but you would
- have gotten 24, and that's larger than necessary.
- You could have still worked it out, but you would have had
- more simplifying left for you to do.
- The same idea we can do right here.
- The same idea we can do here.
- We can factor both of these denominators, so x squared,
- that is x plus 3, times x minus 3.
- It's just a difference of squares.
- And x squared plus 5x, plus 6.
- That is what?
- That is x plus 2, times x plus 3.
- 2 times 3 is 6.
- 2 plus 3 is 5.
- So our common denominator's going to be what?
- What's going to be our common denominator?
- Let me clear up this real estate over here.
- What's our common denominator going to be?
- Our common denominator-- so this is going to be equal to--
- our common denominator's going to be-- our common
- denominator's going to have to have an x plus 3 in it.
- And that takes care of this x plus 3 as well.
- It's going to have an x minus 3 in it.
- It's going to have to have an x minus 3 in it.
- So that's an x minus 3.
- And it's going to have that x plus 2 in it.
- Another way you could have thought about is, like, look,
- it's going to have to have all of the factors of this guy,
- which are x plus 3, times x minus 3. x plus 3
- times x minus 3.
- And then once you write those down, you say, well, it also
- has to have all of the factors for this guy.
- Well, we already have an x plus 3 there, so we just have
- to only add on the x plus 2.
- And that saves us from multiplying both of these
- expressions together.
- This is the least common denominator.
- Now, if this is the least common denominator, what do we
- have to change the 1 by?
- Well, let me actually rewrite this a little bit.
- Well, I'll write it like this.
- So what do you have to do to go from-- if I start with 1
- over x squared-- let me write it this way-- 1 over x plus 3,
- times x minus 3.
- This and this is the exact same thing.
- If I want an x plus 2 in the denominator, what
- do I have to do?
- I have to multiply the numerator and the denominator
- by x plus 2.
- So this expression right here, which is this expression right
- here, will become x plus 2 over x plus 3, times x minus
- 3, times x plus 2.
- Those would cancel out and you're just left with a 1.
- That's that right there.
- So this is the first term.
- That is this term.
- This term over here, the second term over here, this is
- 2 over x plus 2, times x plus 3.
- That's in its original form.
- Now, if we also want an x minus 3 here in the
- denominator, we can multiply it by x minus 3
- over x minus 3.
- Or essentially, we can multiply the
- numerator by x minus 3.
- So this is going to be equal to 2 times x minus 3 is 2x
- minus 6 over x plus 2, times x plus 3, times x minus 3.
- So this expression right here is the exact same thing as
- this expression right over here.
- So if we were to add these two expressions, they have the
- same denominator.
- I just switched the order that I'm multiplying.
- But they're the exact same denominator, which is exactly
- the same thing as this.
- So this, this first expression, we're going to get
- our x plus 2 here.
- x plus 2 over all of this would cancel, and you'd get 1
- over x plus 3, times x minus 3, over 1 over x
- squared minus 9.
- And then to that you're going to add plus 2x minus 6.
- And why am I adding 2x minus 6?
- Because the second term is 2x minus 6 over all of this
- business, because we were just multiplying.
- This would also be written as 2 times x minus 3, over all of
- this. x minus 3's would cancel out and you'd get this again.
- So this is what we get when we add the 2, and so this will be
- equal to-- let me scroll down a little bit-- this will be
- equal to x plus 2x is 3x.
- 2 minus 6 is negative 4.
- All of that over this thing.
- We can either keep it multiplied out, or factor it
- out, or we could multiply it if you like, but it'll take
- some time to multiply it.
- So I'll just say x plus 3, times x minus 3,
- times x plus 2.
- And we are done.