All right in this video. We're going to talk about long division of polynomials and
Long division of polynomials is a lot like normal long division except well now you're doing things with variables
The first thing you want to do for long division of polynomials is basically rewrite it
So that the stuff on the top goes underneath
The bar and whatever you're dividing by goes out front and the tooth a couple things. I want to point out
notice [think] [that] the powers are in descending order it goes from the second power to the first power to just a number and
Likewise the thing you're dividing by it goes in descending powers. We have our first power, and then just a number if
Things aren't written this way. That's the very first thing you'll want to do
Is put them in this order most problems give them to you in descending order?
[but] in case somebody's just trying to confuse you be aware of that
So all you really do is you focus in on the first thing out front here the x and
you focus in on the first thing underneath, and I think well x
X times, what is going to give me x squared well x times?
X is going to give me x times x gives me x squared and then what I do is I multiply
so x times x is x squared
[x] times negative 2 is [negative]
2x and then I put the stuff in parenthesis, and I subtract it away, so I subtract
So x squared minus x squared is no x squared's and that's what always should happen your first thing should cancel out you'll get 4x
minus negative 2x which is 4x plus 2x or
6x and then
We'll drop down our
Negative 8 and
Now I do the same thing as before. I'm still thinking about the x and but now I'm focusing in on the 6x
Once the first term [that] you get after you do your subtraction
Has lesser degree than the thing out front you'll stop so right now x and x are the same degree
So I'm going to do this long division [or] excuse me this multiplication again, so I think x times
what is 6x and I would need a
positive 6 so positive 6 times x is
6x
6 times negative 2 is negative
12
again, I put this in parenthesis and subtract it so
I'll get 6x minus 6x. Which is no x I'll get negative 8 minus negative 12, so that's negative 8 Plus 12
negative 8 plus 12 is Simply 4
So what this says is in conclusion, so this is what we started with x squared plus 4x?
- 8
[x] [-] 2 it says that we can [actually] rewrite that as the stuff on top
x + 6 and then
The 4 out here at the bottom is considered our remainder
so we'll write 4 over what we are originally dividing by
so this would be your
Final solution in terms of rewriting things [ok] so the stuff on top is what goes out front your remainder?
Goes on top of the [fraction] and what you're dividing by goes on the bottom of the fraction
[ok] so let's do maybe one more of these
So in this problem
[ok] same thing notice
I've got things in descending powers [x] cubed x to the first [6] and then x squared [and] z and a plus 1
But notice in the top part, there's no x squared term and in the bottom part
There's no x terms so what I'm going to do is I'm just going to fill that in as a placeholder
So [let] me give myself a little more room [down] here
So I've got x cubed plus 0x
squared
plus 4x plus
6 and
I'm dividing [this] by
x squared
Plus 0 x's
Plus 1 and now I just do the same thing as before
I look at my first term in my first term and I think x squared times, what is x cubed?
well, I would need an x to the first and then I simply multiply everything out front [x] times x is x cubed x
Times 0 x is going to give me well
0 x squared
[x] times 1 is going to give me positive x I
subtract things away, and
In this case I'll get 4x minus x which is
3x and then I'll simply drop my
[+6] down
But now we're already done
There's no more long division to do because if you look at the first term of what I get after I do my subtraction I
get an x to the first term but x squared has a larger degree than x to the first, so
Once that happens once the first term out front has a higher degree than what you get after you do the subtraction
You stop ok so it says basically. We can write x cubed plus
4x Plus 6
Over x squared plus [one] it says we can write that as x and
our remainder
[3] x plus 6
over x squared plus 1 and
Our long division would be complete
ok
So maybe let me do one more because I've got time here, so I'll just make up one more up top of my head
Suppose we have x to the third
plus [two] [x] squared
plus x plus
[1] and we're dividing this all by just again. Maybe a linear term x minus three
So notice now none of the terms are [missing] so I don't have to do
I don't have to fill any of these sort of placeholders in
So I've got x to the third plus [two] [x] squared
plus x plus
One and then outside I'll put my x minus three
so again, I think x times, what is x to the [third] and
that'll be x squared so if I multiply x squared times x is x [to] the third
X squared times negative three is negative
Three x squared
Put this in parentheses and subtract
So the x cubes will cancel out I'll have a [2x] squared plus a 3x squared which is
Five x Squared and I'll drop down my
plus x term and
Now I think x times what is five x squared well, I would need a positive 5x to get that
positive 5x times x is
5x Squared 5x times negative 3 is negative 15 x
So again I do my subtraction
my 5x squared will cancel out I'll get x plus 15 x which is 16 x
and now I drop down my positive 1 we keep going because the thing I'm dividing by the x
has the same degree as my
16 x
So I think x times, what is 16 x?
well in that case I would need a
positive 16
So I'll get positive 16 x 16 times negative 3 is [negative] 48
Again, I put this in parenthesis and subtract the 16 x's will cancel out. I'll get
positive 1 Plus 48 which will give me positive 49
My 49 is just a number that has smaller degree than x so, I'm done
This is going to be my remainder and again what it says is the original thing we were dividing. It's at x cubed
Plus 2x squared plus x plus 1 over
X minus 3 it says you can rewrite this as the stuff on top x squared?
plus 5x plus
16 and then we just put our excuse me put our remainder
49 over what we are originally dividing by which was the x minus 3 and?
that will be your answer after you've done long division, [so]
long division is one of these things it's a little tedious but
it's if if you forget how to do regular division
Regular long division with just numbers I would encourage you to look at that because you're using the same process
so if you've forgotten how to do long division with numbers
[you] [know] you may want [to] review that I think it'll make this stuff
Make some more sense if you have any questions feel free to send me an email
I've got tons of videos on my [website] as well
Dealing with a lot of other Algebra topics too. So feel free to take a look and
Hopefully this helps and like I said if you have any questions, just send me an email, and I'll be happy to answer them