Subtracting rational expressions with factored denominators

- [Voiceover] Pause this video

and see if you can subtract this magenta

rational expression from this yellow one.

Alright, now let's do this together.

And the first thing that jumps out at you

is that you realize these don't

have the same denominator and you would

like them to have the same denominator.

And so you might say,
well, let me rewrite them

so that they have a common denominator.

And a common denominator that will work

will be one that is divisible
by each of these denominators.

So it has all the factors of each

of these denominators and lucky for us,

each of these denominators
are already factored.

So let me just write
the common denominator,

I'll start rewriting
the yellow expression.

So, you have the yellow expression,

actually, let me just make it clear,

I'm going to write both, the yellow one,

and then you're going to
subtract the magenta one.

Whoops. I'm saying yellow
but drawing in magenta.

So you have the yellow expression

which I'm about to rewrite, actually,

I'm going to make a longer line,

so the yellow expression
minus the magenta one,

minus the magenta one, right over there.

Now, as I mentioned, we
want to have a denominator

that has all, the common
denominator has to have,

be divisible by both,
this yellow denominator

and this magenta one.

So it's got to have
the Z plus eight in it.

It's got to have the 9z minus five in it.

And it's also got to have both of these.

Well, I already, we already
accounted for the 9z minus five.

So it has to have, be divisible
by Z plus six. Z plus six.

Notice just by multiplying the denominator

by Z plus six, we're not
divisible by both of these factors

AND both of these factors
because 9z minus five

was the factor common to both of them.

And if you were just dealing with numbers

when you were just adding
or subtracting fractions,

it works the exact same way.

Alright, so what will
the numerator become?

Well, we multiply the denominator times

Z plus six, so we have to do the

same thing to the numerator.

It's going to be negative Z
to the third times Z plus six.

Now let's focus over here.

We had, well, we want
the same denominator,

so we can write this as Z plus eight

Z plus eight times Z plus six,

times Z plus six times 9z minus five.

And these are equivalent.

I've just changed the order
that we multiply in it,

that doesn't change their value.

And if we multiplied the, so
we had a three on top before

and if we multiply the
denominator times Z plus eight,

we also have to multiply the
numerator times Z plus eight.

So there you go.

And so, this is going to be equal to,

this is going to be equal
to, actually, I'll just make

a big line right over here.

This is all going to be equal to.

We have our, probably
don't need that much space,

let me see, maybe that,
maybe about that much.

So I'm going to have the same denominator

and I'll just write it
in a neutral color now.

Z plus eight times 9z minus
five times Z plus six.

So over here, just in this blue color,

we want to distribute this
negative Z to the third.

Negative Z to the third times
Z is negative Z to the fourth.

Negative Z to the third times
six is minus 6z to the third.

And now this negative
sign, right over here,

actually, instead of saying negative Z,

negative of this entire thing,

we could just say plus
the negative of this.

Or another of thinking about it,

you could view this as negative
three times Z plus eight.

So we could just distribute that.

So let's do that.

So negative three times Z is negative 3z

and negative three times
eight is negative 24.

And there you go.

We are, we are done.

We found a common denominator.

And once you have a common denominator,

you could just subtract
or add the numerators,

and instead of doing this
as minus this entire thing,

I viewed it as adding and then having

a negative three in the numerator,

distributing that and then these,

I can't simplify it any further.

Sometimes you'll do one of these types

of exercises and you might have

two second-degree terms
or two first-degree terms

or two constants or something like that

and then you might want
to add or subtract them

to simplify it but here, these
all have different degrees

so I can't simplify it any
further and so we are all done.