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!