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# Multiplying and Dividing Rational Expressions

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- Let's do some more work with rational expressions.
- So let's say I had a squared plus 2ab plus b squared and
- all of that is over ab squared minus a squared b.
- And we're going to take this and divide it by a plus b.
- So the first thing we might want to do is just factor this
- numerator and this denominator, and
- then we could divide.
- Or actually, we could go the other way.
- We could divide and then factor.
- So if we divide and then factor, we could say, well,
- this is the same thing.
- This is equal to this whole expression.
- Let me just copy and paste it.
- That's probably the easiest thing to do.
- Copy and then paste it.
- It's equal to that expression times the inverse of this.
- If I divide by something, that's the same thing as
- multiplying by the inverse, as multiplying by
- 1 over a plus b.
- And this is just going to be the same thing as that with a
- plus b in the denominator, because your numerator's going
- to be that times 1, which is just that.
- Your denominator is going to be ab squared minus a squared
- b times a plus b, so that times a plus b.
- So this would be a legitimate answer, but I have a suspicion
- that this can be further simplified.
- So let's see if we can simplify it.
- So our numerator, a squared plus 2 ab plus b squared, you
- might recognize that.
- That is a perfect square.
- That is our numerator.
- Let me color code it.
- So that numerator right there, that is the same thing as a
- plus b squared, or a plus b times a plus b, because you
- have a squared, you have a b squared, and then you have 2
- times a and b right in the middle.
- That is a perfect square.
- Multiply it out if you don't believe me.
- So that numerator is that right there.
- And then the denominator, what is this?
- Well, what happens if we factor out an ab down here?
- Let's factor out an ab.
- So ab squared divided by ab, a cancels out, you're just left
- with ab, right? ab times b is ab squared, and then minus a
- squared b divided by ab is just a.
- This right here is the same thing as that.
- And you can multiply it out. ab times negative a is
- negative a squared b.
- You got that right there.
- So multiply it out.
- You should get this right there.
- And, of course, you also have your a plus b sitting here.
- Now, we have an a plus b squared in the numerator, we
- have an a plus b into the denominator, or we could say a
- plus b to the first power.
- And then we know our exponent rules here.
- We can just essentially cancel out an a plus b in the
- numerator and the denominator, which decrements this exponent
- by 1, so this just becomes a 1.
- And then this becomes a zero, or it cancels out.
- And so we get this is being equal to a plus b over ab
- times b minus a with the caveat because we canceled
- this thing out in the denominator.
- We have to say, look, this thing could still not equal
- zero, because this will still make
- this expression undefined.
- We have to say the only way that this is going to be equal
- to zero is if we subtract b from both sides, a is equal to
- negative b.
- So we have to add the caveat that a cannot be equal to
- negative b.
- We have to add this condition.
- So this is our answer.
- Let's do another one.
- Let's say I have-- let's do a more involved one, just to
- really get our juices flowing.
- Let's say we have x squared plus 8x plus 16 divided by, or
- over, 7x squared plus 9x plus 2 divided by 7x over 4 over x
- squared plus 4x.
- So what is this going to be equal to?
- So here we could divide first. That's the same thing as
- multiplying by the inverse of this thing.
- But just for fun, let's factor these two, the numerator and
- the denominator here.
- So this numerator over here, that's pretty straightforward.
- That is a perfect square.
- That is x plus 4 squared.
- 4 times 4 is 16.
- 4 plus 4 is 8.
- So our numerator is x plus 4 squared.
- And then our denominator, what does that become?
- Here, we're going to have to do a little bit of grouping to
- factor that out.
- So we have 7x squared plus 9x plus 2.
- We have to find two numbers.
- When I multiply them, I get 7 times 2, which is equal to 14.
- And when I add them, a plus b, they equal 9.
- And the easiest two numbers are 7 and 2.
- So let's rewrite this.
- This is the same thing as 7x squared-- I'll group the 7x
- with the 7 because they have a common factor of 7-- plus 7x
- plus 2x plus 2.
- I put parentheses around there so you can see the grouping.
- Factor out a 7x over here, you get 7x times x plus 1 plus--
- factor out a 2 here.
- 2 times x plus 1, and you get 7x plus 2 times your x plus 1.
- We're undistributing that expression.
- So this is our denominator right there.
- And actually, I just realized that I made a slight error
- when I wrote down this problem.
- The numerator here is actually 7x plus 2, lucky for us.
- Obviously, this is going to lead to interesting things
- later on, potentially.
- So let's see what we get.
- And actually, I think there's a typo here.
- I think this is supposed to be a 4x-- oh, sorry, just a 4.
- So let me just put it like that just in case that is a
- typo because I think it's going to make it more
- interesting.
- So that's our original problem.
- We haven't messed with this part yet, so it won't change
- any of our math.
- But so far, we factored this bottom part.
- We factored this bottom part using grouping as 7x plus 2
- times x plus 1.
- And so this expression is the exact same thing as this
- expression over here, and we're going to divide it by
- that expression.
- So dividing by that expression, that is the same
- exact thing as multiplying by the inverse of this
- expression.
- Actually, you know what?
- I just realized that it wasn't x squared plus 4x.
- I just saw the problem.
- That is an x squared plus 4x.
- This was the original problem.
- Sorry for all that.
- We haven't done any work here so it's all cool.
- So this right over here, multiplying by the inverse of
- this, we multiply-- dividing by this is the same thing as
- multiplying by the inverse, so we just flip the numerator and
- the denominator.
- So x squared plus 4x over 7x plus 2.
- And then another thing we can do, if we look at this up
- here, we can factor out an x.
- So this right here, if we factor out an x is the same
- thing as x times x plus 4.
- So if we do all of the multiplication, what does our
- numerator become?
- We have x-- I'll use a new color-- we have x plus 4
- squared times x times x plus 4.
- That is our numerator.
- And our denominator is 7x plus 2 times another 7x plus 2
- times an x plus 1.
- So here nothing is canceling out, but we have the same term
- being multiplied multiple times.
- x to the fourth squared times x to the fourth to the first
- power, this is going to be equal to-- we still have our x
- out front, so it's x times x plus 4, we could say to the
- third power, all of that over-- we have two 7x's plus
- 2, so 7x plus 2 squared times x plus 1.
- And we are done!

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