>> In this video we're going to go over how to divide a polynomial by a monomial.
So remember a monomial is a single term.
So, for example, you might have something
like 3x cubed plus 12x squared minus 9x divided by 3x.
The numerator is a polynomial.
It could -- It's got more than one term; it happens to be a trinomial.
But in the denominator, there's only a single term, 3x.
So this would be a monomial.
And to do this, we simply break up the, the numerator in this fashion
and put each term over the monomial.
So that's 3x cubed over 3x plus 12x squared over 3x minus 9x over 3x.
And you should convince yourself that we're really just using the rule
for adding and subtracting fractions.
Remember, if you add, if you have a common denominator you simply add the numerators
together, so you can see that if you went backwards, if you were to add this,
since they all have a common denominator, you would end up right where we started.
So we're actually kind of going backwards.
Now each of these we've already gone over: how do you divide a monomial by a monomial?
So now you just cancel.
Now in here the threes cancel, and then you've got an x cubed over x to the first,
so you could either use your laws of exponents and say that's x
to the 3 minus one, which is x squared.
Or you could actually write out x times x times x all over x,
and show that one of the x's cancels.
Okay. Onto the second, it's going to cancel.
And we have x squared over x, so that's going to give us four, and it's just going
to be one x. Again remember that's x squared over x is like over x to the first power.
And onto the third one.
The three goes into the nine, and also both x's cancel.
So this is just going to be minus 3.
And that's your answer.
So that's our answer.
[ PAUSE ]
Now sometimes when you do this, there might have been another term that didn't cancel completely.
So the last term here might not have just been a minus 3; you might have had another fraction.
So let's do one like that now.
Let's say we have 12x cubed y squared minus 15x y to the fourth plus -- let's do 20x squared.
And let's say we were dividing this by 4x squared y. Okay,
so this is going to look a little bit different.
Whoops. Sorry, that disappeared for a second.
So we're going to have 12x cubed y squared over 4x squared y. And we go to the next term:
15xy to the fourth over 4x squared y. And 20x squared over 4x squared y.
And we're back to canceling as much as we can.
Alright, so again you could use the laws of exponents here, or you could actually, you know,
completely factor everything and cancel everything out.
That takes more space.
I'm not going to do it, so I can cancel the 12 and the 4.
That's going to give you a 3.
And remember that's a 1 at the bottom.
And let's see, with the x cubed over the x squared, that's if we subtract exponents,
we're just going to end up with an x at the top, right?
And the y squared over the y -- that's going to end up with a y. So for this first term,
it all canceled out and we got just ones in the denominator, so we're going to have 3xy.
Let's go on to the next part.
Well, there's no common factors between 15 and 4,
so we're not going to be reducing that at all.
Let's see, x over x squared.
There's more x's in the bottom, right?
You have x times x, so I could cancel here and I'm going
to be left with just one x in the bottom.
And with a y to the fourth over y, you're going to have to subtract exponents.
That's going to give you y cubed in the numerator.
So this leaves me with 15y cubed in the numerator over 4x in the denominator.
And if we go over to the third terms over here, 20 over 4 reduces completely.
The x squared over x squared reduces completely.
So I have just a 5 -- 5 in the numerator and a y in the denominator.
So just keep in mind that when you're dividing by a monomial, you might end up with something
like this where you have a fraction, in this case in two parts.
Alright, here's our answer, my answer.
So it really just depends, but the main thing is that first you just break up the numerator
and put it over each denominator.
Alright, here's one more example.
In this case, there's only two terms in the numerator,
but again we're just dividing by a monomial, a single term.
But you've got to be very careful with the negative signs here.
Alright, so let's do this one.
We have 9x to the fourth over a negative 3x squared.
And here I've got a minus 15x cubed over a negative 3x squared.
Alright, now because you have minus sign and then another minus sign in this case
that could be changed to plus and plus.
But here we've got a positive divided by a negative.
So we are going to have to deal with that.
So let's see, try it.
Put it on pause and try it.
We're back.
So 9 divided by negative 3 is negative 3.
x to the fourth divided by x squared; you can subtract exponents
and that will give you x squared.
And 15 divided by 3 is 5, and x cubed divided by x squared --
if we subtract exponents that's actually the first
which will just be x, and so this is your answer.
I'm going to do one -- this the long way just as a little review.
If you forget your laws of exponents, you can go ahead and cancel the long way.
So let's do this.
Let's take the same problem.
Alright, starting out the same way, I could write 9x to the fourth
over negative 3x squared minus 15x cubed over negative 3x squared,
but then watch what I'm going to do from here.
I'm going to simply write out -- I'm going to make this a little smaller.
Here we go.
Okay. I'm going to show the total factors here.
So this 9x to the fourth, remember, is 3 times 3 -- I'm prime factoring --
times x times x times x times x. That's x to the fourth.
And in the denominator, this negative 3 I could just leave as negative 3, or I could think of it
as negative 1 times 3, which I'm going to do.
And x times x. That's just the first term over the second term,
so it's the first term in this binomial.
And I'm still going to deal with this plus and plus right now,
so we don't have to worry about that.
But 15, remember, is 3 times 5, and x cubed is x times x times x.
And in the denominator I have 3 times x times x. Now if you do this, it's sort of fun.
You just gotta cancel everything out that's exactly the same.
That x and that x. That x and that x. This 3 and that 3.
Okay, let's go over here.
I could cancel the 3 with that 3.
This x with that x. This x with that x. Yeah, so now what do I end up with?
Here I've got the 3, the x and the x, so you could -- sometimes it's a good idea to, like,
circle it so you don't miss something.
You've got all of that in the numerator divided by all of that.
Now you never leave a negative 1 or a positive 1 in the denominator.
If it's a negative 1, it means you just change the sign of the numerator,
and if it's a positive 1 you can just leave it off completely.
So this gives you negative 3x squared.
And over here what do we have left?
Alright, I've got the 5 and the x, and over here I don't have anything in the denominator,
but remember you might not see it, it's as if there's still a 1.
So it's not like there's a 0, but there is something.
But it's a 1 and you don't write the 1.
So when we simplify it, we're just going to get plus 5x.
Alright, now is that the same thing we got
when we did it the other way using the laws of exponents?
Negative 3x squared plus 5x?
Yes, there it is: negative 3x squared plus 5x.
Negative 3x squared plus 5x.