Simplify the rational expression and state the domain.
So we have a rational expression here.
One expression being divided by another one.
And so to simplify it, I am guessing
that we will have to factor this top expression
and this bottom expression.
And there's probably common factors
that we might be able to cancel out.
But we have to be very careful about how we do them.
So let's factor each of these.
So you may or may not remember how
to factor quadratic expressions that
have a non-one coefficient out here.
So this will be practice in that.
And we go into the rationale for why it works in those videos
where we first learned to factor rational expressions.
And they're in this playlist.
But just as a reminder, what you want to do is you
want to split this middle term.
You want to split the first degree term into two terms.
You want to group it, one of them with this 2x squared
and one of it with the 20.
And you want to group it into two terms
so that the coefficients of the two terms,
when you take their product, is the same thing as 2 times 20.
So you want to break-- let me write it over here.
You want to break 13x, you want to break it
into an ax plus a bx, where a times b is equal to 2 times 20.
And we show why this works in a previous video.
So a times b needs to be equal to 40.
And a plus b needs to be equal to 13.
So let's just think about the factors of 40.
1 and 40 are not going to work.
2 and 20 are not going to work, they don't add up to 13.
3 and 3-- well, 3 doesn't go into 40.
4 and 10 does not work.
5 and 8-- so if we take 5 times 8, that does equal 40.
And then 5 plus 8 does equal 13.
So that's looking pretty good.
So let's split up the 13x along those lines.
So we get 2x squared plus 5 plus 5x plus 8x is plus 20.
So all I did is I split up this 13x into a 5x plus an 8x.
And now we can group these two terms.
And then let's go down over here and do the same thing
in the denominator.
So let's do the same exact thing in the denominator.
Once again, we want to find two numbers that add up to 17
and whose product is 2 times 30, or 60.
So if we look at the factors of 60, 1 and 60 don't work.
They don't add up to 17.
Or if you took the negative 1 and negative 60--
you're not going to have negative numbers because they
would have to be the same sign since this is positive.
And then they would give you a negative value over here.
So we have to deal with positive numbers.
2 and 30 don't work.
3 and 15-- remember, we want to get to 2 times 30,
we want to get to 60.
3 times 15-- sorry, not 3 times 15.
3 times 20 doesn't work.
Their product is 60, but it doesn't add up to 17.
4 times 15 does not work.
5 times 12-- 5 times 12 is 60, so that works.
That's the same thing as 2 times 30.
It equals 60.
And 5 plus 12 is 17.
So that's looking pretty good.
So let's separate this out into 2x
squared plus 5x plus 12x plus 30.
And then we can group them like this.
And then let's see how we can simplify
each of these expressions.
This top expression over here, we have a common factor of x.
So this top expression over here,
we can factor out an x out of this.
So this becomes x times-- and 2x squared divided by x is 2x.
5x divided by x, when you have the x factored out, that
is just 5.
So this first term is x times 2x plus 5.
And then plus-- over here we can factor out a 4.
4 is divisible into both 8x and into 20.
So let's factor out a 4.
So this is 4 times-- 8x with a 4 factored out is now just 2x.
And then 20 with a 4 factored out is just a 5.
And then let's do the bottom over here.
I'm kind of doing them both simultaneously in parallel.
The bottom over here, we can factor out
an x out of this expression.
So this is x times-- once again, 2x plus 5.
Plus, this expression over here, we can factor out a 6
out of both 12x and 30.
So let's factor out a 6.
And then 12x divided by 6 is 2x.
And then 30 divided by 6 is 5.
2x plus 5.
And so we can simplify this.
We can factor out, from this term
we can factor out a 2x plus 5, from this term we
can factor out a 2x plus 5.
So if we factor out a 2x plus 5 from both of these--
so we're factoring out a 2x plus 5,
you're left with an x and a plus 4.
You're left with an x plus 4.
And then down here we can factor out a 2x plus 5.
Down here we can factor out a 2x--
let me make it in that same color.
We can factor out a 2x plus 5.
So we're literally just factoring out a 2x plus 5,
and we're left with an x plus 6.
We're just doing the distributive property
in reverse over here. x plus 6.
Now, you might be tempted to say,
hey look, I have a 2x plus 5 in the numerator, 2x
plus 5 in the denominator, let me just cancel them out
really fast and I would have simplified the expression.
We have to be very careful, because we first
need to think about the domain.
Because if we just simplify this expression,
and we include things in the simplified domain that
weren't included in the original function's domain,
we've actually changed the expression.
Or we've changed the function.
So we have to be very careful that even when we simplify it,
we keep the domain the same.
So what's the domain of this thing?
Well, it's going to be all real numbers except the ones that
make the denominator equal to 0.
And what are the numbers that make
the denominator equal to 0?
Well, the denominator is equal to 0 if x plus 6 is equal to 0,
or 2x plus 5 is equal to 0.
And if x plus 6 is 0, then that means that x is negative 6.
Subtract 6 from both sides.
Over here we could subtract 5 from both sides.
So you get 2x is equal to negative 5.
Divide both sides by 2, you get x is equal to negative 5/2.
So the domain of this expression right
over here, the domain of this expression-- if we could call
this a function, if we defined it
as f of x is equal to all of this-- the domain of it
is going to be all real numbers, except negative 6
and negative 5/2.
These will make this expression undefined.
It will make this denominator equal to 0.
Now we can simplify it, as long as we keep this denominator.
So now, if we assume that we're never
going to see a negative 5/2, then 2x plus 5
is never going to be equal to 0, and so we
can divide the numerator and denominator by that value,
canceling it out.
And so the simplified expression is what's left over.
x plus 4 over x plus 6.
But we have to keep the same domain.
If we didn't write this here, you'd
say, OK, the only thing that would
make this thing 0 is x is equal to negative 6.
We have to remember, this is a simplified version of that.
And in order for these to be identical expressions, or maybe
identical function definitions, they
need to have the exact same domain.
So the domain for this simplified expression
has to be the same as the original.
All real numbers except negative 6 and negative 5/2.