>> Let's talk about some common errors dealing
with rational expressions.
First thing you need to make sure that you do is
that you only cancel like-factors.
Okay? So that you can only cancel like-factors.
Let's say you had a problem that looked like this.
3X times 5 over, let's say, 6.
Notice that everything on the top is being multiplied.
So I can cancel 3 in the numerator.
3 divides into the numerator one time.
3 divides into the denominator 2 times,
which leaves me with 5X over 2.
So I divided the numerator by 3
and I divide the denominator by 3.
I canceled like-factors.
Everything on the numerator was being multiplied.
They were like-factors.
Everything was being multiplied.
Factors are things that are being multiplied.
Look at my first problem.
On my first problem a very common error is for people
to cancel the negatives, but is that negative
in the numerator a factor of everything in the numerator?
Oh no it is not.
Let's make it one, though.
In other words, let's factor the negative
out of everything in the numerator.
So let's factor a negative out.
So that negative-6X -- we're going to factor a negative
out so that becomes a positive-6X.
That plus 7 now will become subtract 7.
Now we cancel those negatives.
On the outside there we cancel the negatives.
We cancel a negative-1 on top.
Cancel a negative on the bottom.
And now we get 6X minus 7 over positive-5.
Notice that's different than what would have happened
if I would have canceled the negatives on the outside
on the original problem.
On our second problem it looks
like you can cancel the 6 with a 3.
That's the most common error.
You cannot.
If instead of subtraction that was multiplication,
you would be able to cancel it.
So how do we do this problem?
We circle it and say we're finished.
There's no way to simplify the problem.
You're done!
On this problem it also looks
like you can cancel those negatives out front.
Once again, that's the most common error.
No, you can't do that.
So like on we did on the first problem, you could factor
out the negative on the front -- on the top.
So factor out a negative.
It was a negative-X, now it's positive.
It was subtract 5, now it's plus 5.
Now we cancel a negative-1 on the top, which is a factor
of the numerator, and a negative-1 on the bottom,
which is a factor of negative-4, which makes it positive-4,
which gives us our answer.
But let me show you another way to do the same problem.
We could multiply the numerator by negative-1
and the denominator by negative-1.
Now what's that fraction equal to?
Ah. It's equal to the number 1.
You can multiply anything you want by the number 1, right,
that doesn't change it.
That's legal.
You can multiply anything you want by the number 1.
What does that give us?
That gives us positive-X plus 5 over positive-4.
There we go.
Same thing.
For our next problem -- students
like to cancel all kinds of things.
They cancel, I don't know, the X-squares,
and they cancel the X's and, you know,
just kind of anything they want to.
Well, what gives them the right to cancel those?
Are those factors?
Is the numerator factored?
Is the denominator factored?
No, it's not factored.
Nothing's factored!
In fact, nothing on their factors at all.
What's the answer to that problem?
That. Now if it did factor then you would factor it,
and then if like-factors cancel then you would do that.
But right now nothing is factored
so you cannot cancel anything.
On our last problem we're going to cancel the X plus 3s, right?
Look at the denominator, it's all nice and factored.
We know it's factored.
It's gorgeous.
So we're going to cancel the X plus 3s, right?
Yeah, yeah, okay.
Cancel the X plus 3s?
Right, right, yeah?
Wait a minute.
Look at the numerator.
Is that a multiplication symbol?
Uh-oh, wait a minute.
If that's not a multiplication symbol
that means the numerator is not factored.
If the numerator is not factored I can't cancel anything.
I cannot cancel the X plus 3s.
I cannot cancel the X minus 7s.
I cannot cancel anything.
Boo.
So, distribute but leave the denominator alone.
We love the denominator.
It's factored.
It's beautiful.
Leave it alone.
All done.