Adding and subtracting rational expressions with like denominators

- [Voiceover] So let's add

six over two X squared minus seven

to negative 3 X minus eight

over two X squared minus seven.

And like always, pause the video

and try to work it out before I do.

When you look at this,

we have these two rational expressions

and we have the same denominator,

two X squared minus seven.

So you could say, we have six
two X squared minus sevenths

and then we have negative
three X minus eight

two X squared minus sevenths
is one way to think about it.

So if you have the same denominator,

this is going to be equal to,

this is going to be equal to...

our denominator is going to
be two X squared minus seven,

two X squared minus seven,

and then we just add the numerators.

So it's going to be six
plus negative three X,

negative three X minus eight.

So if we want to simplify
this a little bit,

we'd recognize that we can
add these two constant terms,

the six and the negative eight.

Six plus negative eight is going to be

negative two, so it's
going to be negative two

and then adding a negative three X,

that's the same thing
as subtracting three X,

so negative two minus
three X, all of that over,

all of that with that same blue color,

all of that over

two X squared minus seven.

And we're done.

We've just added these
two rational expressions.

Let's do another example.

So here, we want to subtract

one rational expression from another.

So see if you can figure that out.

Well, once again, both of
these rational expressions

have the exact same denominator,

the denominator for both of them is

14 X squared minus nine,
14 X squared minus nine.

So the denominator of the difference,

I guess we can call it that,

is going to be 14 X squared minus nine.

So 14 X squared minus nine.

Did I say four X squared before?

14 X squared minus nine,

that's the denominator of both of them,

so that's going to be the denominator

of our answer right over here.

And so, we can just
subtract the numerators.

So we're gonna have

nine X squared plus three

minus all of this business,

minus negative three X squared plus five.

And so we can distribute
the negative sign.

This is going to be equal to

nine X squared plus three, and then,

if you distribute the negative sign,

the negative of negative three X squared

is going to be plus three X squared

and then the negative of positive five

is going to be negative five,

so we're gonna subtract five from that,

and all of that is going to be
over 14 X squared minus nine.

14 X squared minus nine.

And so in the numerator we
can do some simplification.

We have nine X squared
plus three X squared,

so that's going to be
equal to 12 X squared.

And then, we have...

we have three plus negative five,

or we can say three minus five,

so that's going to be negative two,

and all of that is going to be over

14 X squared minus nine.

14 X squared minus nine.

And we're all done.

We have just subtracted.

And we can think about it,

is there any way we
can simplify this more,

are there any common factors,

but these both could be
considered differences of squares,

but they're going to be
differences of squares

of different things,

so they're not going
to have common factors.

So this is about as simple as we can get.