- [Voiceover] So I have a rational expression here
and my goal is to simplify it, but while I simplify it,
I wanna make the simplified expression
be algebraically equivalent.
So, if there are certain x values
that would make this thing undefined,
that I have to restrict my simplified expression
by those x values.
So, you could pause this video and take a go at it
and then we'll do it together.
All right. So let's just think real quick.
What x values would make this expression undefined?
Well, it's undefined if we try to divide by zero.
So if x is zero, 14 times zero is zero,
it's going to be undefined.
And so, we could say this is,
we could say x does not, cannot be equal to zero.
For any other x, we can evaluate this expression.
Now let's actually try to simplify it.
So, when you look at the numerator and the denominator,
every term we see is divisible by x
and every term is divisible by seven.
So it looks like we can factor seven x out of the numerator
and seven x out of the denominator.
So the numerator, we can rewrite
as seven x times,
we factor seven x out of 14 x squared,
we're gonna be left with two x
and then you factor seven x out of seven x,
you're gonna be left with a one.
And one way to think about it
is we did the distributive property in reverse,
if we were to do it again,
seven x times two x is 14 x squared,
seven x times one is seven x.
All right. And now let's factor seven x
out of the denominator.
So, 14 x could be rewritten
as seven x times two,
times two,
and remember, I wanna keep this algebraically equivalent,
so I wanna keep the constraint
that x cannot be equal to zero.
And so we divide the numerator
and the denominator by seven x or one way to think about it,
you can divide seven x by seven x and just get one
and we are left with
two x plus one over two.
Now this was the original expression,
x could take on any value,
but if we want it to be algebraically equivalent
to our original expression,
it has to have the same constraints on.
So we're going to have to say x,
x does not equal zero,
and it's a very subtle but really important thing.
For example, if you defined a function
by this right over here,
the domain of the function could not include zero.
And so, if you simplified how you defined
that function to this,
if you want that function to be the same,
it needs to have the same domain.
It has to be defined for the same inputs.
And so that's why we're putting
the exact same constraints for them to be equivalent.
If you got rid of this constraint,
these two would be equivalent everywhere
except for x equal zero.
This one would have been defined for x equal zero,
this one would not have,
and so they wouldn't have been algebraically equivalent.
This makes them algebraically equivalent.
And of course you can write this in different ways.
You can divide each of these terms by two, if you like.
So you could divide two x by two and get x
and then divide one by two and get 1/2.
Once again, we would wanna keep
the x cannot be equal to zero.
Let's do another one of these.
So it's a slightly hairier expression,
but let's do the same drill.
See if we could simplify it.
But as we simplify it, we really wanna be conscientious
of restricting the Zs here
so that we get an algebraically equivalent expression.
So, let's think about where this is undefined.
We can think about where it's undefined
by factoring the denominator here.
Actually, let me just...
So, this is going to be equal to,
actually, let me just...
Let me just do the first step where I could say,
"Well, what's a common factor
"in the numerator and the denominator?"
Every term here is divided by z squared
and every term is also divided by 17,
so it looks like 17 z squared can be factored out.
So, 17 z squared can be factored out of the numerator
and then we would be left with,
we factor out of 17 z squared out of 17 z to the third
and we're gonna be left with just a z.
17 z squared, you factor 17 z squared out of that
and you're gonna be left with just a one.
And once again, you can distribute the 17 z squared,
multiply it times z, you get 17 z to the third.
17 z squared times one is 17 z squared.
All right. All of that is going to be over,
we wanna factor out of 17 z squared
out of the denominator,
17 z squared times,
and so 34 z to the third divided by 17 z squared,
34 divided by 17 is two,
and z to the third divided by z squared is z
and then we have minus 51 divided 17 is three
and z squared divided by z squared is just one.
So we'll just leave it like that.
And so over here, it becomes a little bit clear
of, well one, how we're gonna simplify it.
We're just gonna divided 17 z squared by 17 z squared,
but let's be careful to restrict the domain here.
So, we can tell if z is equal to zero,
then this, 17 z squared is gonna be equal to zero
and we make the denominator equal to zero
and we could see that even looking here.
So we could say z cannot be equal to zero.
Now what else? Z cannot be equal to
whatever makes this two z minus three equals zero.
So let's think about what makes
two z minus three equal to zero.
Two z minus three is equal to zero.
You can add three to both sides
and you would get two z is equal to three.
Divide both sides by two
and you would get z is equal to 3/2.
So z cannot be equal to zero
and z cannot be equal to 3/2.
So that's how we're gonna restrict our domain.
But now let's simplify.
So, if we simplify it, these two cancel out
and we are going to be left with,
we're gonna be left with z plus one
over two z minus three
and we wanna keep that constraint.
z cannot be equal to zero.
And actually, this second constraint is redundant
because we still have the two z minus three here.
If someone were to just look at this expression,
like, well, the denominator can't be equal to zero
and so, z cannot be equal to 3/2.
So this is still, if we just left it the way it is,
we don't even have to rewrite this,
that would be redundant since looking at this expression
is clearly not defined for z equals 3/2.
So, there you go.
And so I'm gonna ask you
for what values does this expression not defined,
well then you would also include that.
Actually, let me just write it.
It doesn't hurt to be redundant.
z does not equal 3/2.
But this constraint right here is really important
because it's not obvious by looking at this expression.
This expression by itself
would be defined for z equals zero,
but if we want it to be algebraically equivalent
to this one and that one,
it has to be constrained
in the same way.