Adding rational expression example

- [Voiceover] Pause the
video and try to add

these two rational expressions.

Okay, I'm assuming you've had a go at it.

Now we can work through this together.

So the first thing that you might have hit

when you tried to do it,

is you realized that they
have different denominators

and it's hard to add fractions

when they have different denominators.

You need to rewrite them

so that you have a common denominator.

And the easiest way to
get a common denominator

is you can just multiply
the two denominators,

especially in case like this

where they don't seem
to share any factors.

Both of these are about
as factor as you can get

and they don't share anything in common.

And so let's set up a common denominator.

So this is going to be equal to

it's going to be equal to something

let's see, it's going
to be equal to something

over our common denominator.

Let's make it

let's make it 2x,

I'm going to do this in another color.

So we're going to make it 2x-3

times 3x+1

times 3x+1

and then plus

plus something else

over 2x-3

2x-3

times 3x+1.

Times 3x+1.

And so to go from 2x,

to go from just a 2x-3
here the denominator

to a (2x-3)(3x+1)

we multiply the denominator by 3x+1.

So if we do that to the denominator,

we don't want to change the value

of the rational expression.

We'd also have to do
that to the numerator.

So the original numerator was 5x.

I'll do that in blue color.

So the original numerator was 5x

and now we're going to
multiply it by the 3x+1,

so times 3x+1.

Notice I didn't change the
value of this expression.

I multiplied it by 3x+1 over 3x+1,

which is 1 as long as
3x+1 does not equal zero.

So let's do the same thing over here.

Over here I have a denominator of 3x+1,

I multiplied it by 2x-3,

so I would take my numerator,

which is -4x²

and I would also multiply it by 2x-3.

2x-3.

Let me put parentheses around this

so it doesn't look like
I'm subtracting 4x².

And so then I can rewrite
all of this business

as being equal to,

well, in the numerator,

in the numerator I'm
going to have 5x times 3x

which is 15x²

5x times 1, which is + 5x,

and then over here,

let me do this in green,

let's see, I could do -4x times 2x

which would be -8x²

and then -4x times -3

which is +12x².

Did I do that right?

Negative...

Oh, let me be very careful.

- 4, my spider sense could tell
that I did something shady.

In fact, if you want to pause
the video you could see,

try to figure out what
I just did that's wrong.

So -4x² times 2x

is -8x to the third power.

- 8x³

and then -4x² times -3 is 12x²

and then our entire denominator

our entire denominator

we have a common denominator now

so we were able to just add everything

is 2x-3

2x-3

times 3x+1

times 3x+1

and let's see, how can we simplify this?

So this is all going to be equal to,

let me draw,

make sure we recognize
it's a rational expression,

and so let's see, we can look at, we can,

our highest degree term here is the -8x³

so it's -8

- 8x³

and then we have a 15x²

and we also have a 12x².

We could add those two together

to get a 27x²

so we've already taken care of this,

we've taken care, let me
do it in that green color,

so we've taken of this,

we've taken care of those two
and we're just left with a 5x,

so + 5x

and then all of that is over

2x-3 times 3x+1

3x+1

and we are

and we are all done.

It doesn't seem like there's any easy way

to simplify this further.

You could factor out an
x out of the numerator

but that's not going to cancel out

with anything in the denominator

and it looks like we are all done.