Hello everyone.
In this video, we're going to talk about long division,
and we're going to do it over a couple of different fields.
So what are the quotient and remainder when
this quintic polynomial is divided by this quadratic polynomial
over the real numbers?
And what if we do the computations over Z 5, Z mod 5?
What changes in this situation?
Okay, so instead of typing this out, I'm actually going to do this
on paintbrush.
So let's bring it up. Okay,
so here I've written down the question again.
We'd like to long divide these two polynomials.
As you've noticed here in the middle,
I have this extra x cubed term with a 0 coefficient.
It's just a placeholder to help me keep everything organized as I attempt to go through this.
Okay, so let's begin.
The question that I want to know is what number do I have to multiply x squared by to get to x to the 5
over R? So this computation is going to be over R, maybe I'll put a little R over here.
Just to help remind us.
Okay, so what number do I have to multiply x squared by to get to x 5,
or what object, I guess, it's not really a number.
x cubed.
Now I take x cubed, multiply it by this whole polynomial,
what do I get? I'm going to get x to the 5.
If these don't match up, then you've made a mistake up here,
and I'm going to get minus 2x
to the 3 plus 1 which is 4.
Now I subtract,
and that's going to give me 5x to the 4.
Now I'm going to bring down the 0 term,
again it's not important that you write this. I'm just doing it for
organizational purposes.
x squared, now the number I'm going to multiply to get to 5x to the 4 that's 5x squared.
So that's going to give me
5x to the 4,
and then I'm going to take 5x squared multiply by the minus 2x,
that’s going to be minus 10x cubed.
Now we subtract.
First two terms cancel again. Next two terms, it's going to give me a 10x cubed.
Okay?
Plus, let's bring down the next term, 2x squared.
x squared multiplied by 10x is 10x cubed,
so up top I go 10x…
by the way, another way to see this, doesn't always work -
well it does always work we have to think of the word divide carefully,
you could divide 10x cubed by x squared,
and that will give you the number up there.
You have to divide properly in whatever field you're working in.
In this case, it's R, so we have a good understanding of what's going on.
10x cubed
and minus 20x squared,
so it should be a 2.
Subtract,
so it's going to be 22
x squared.
I'm going to drop down the 4x,
I'm going to do the long division so it goes in 22 times.
Okay? So I'm going to get
22x squared
minus, what is that, 44x.
And my last subtraction
is going to give me 48x
plus 1.
And remember that this is the remainder,
the 48x plus 1, so maybe I'll write the word remainder.
So that's my remainder, and up top is my quotient.
Okay, great. So there's a long division example over R, okay?
Now what I'd like to do is I'd like to do the same computation over Z mod 5
and see what changes, okay?
A lot of - okay, and we'll see what changes.
Over Z mod 5.
Okay, so let's see what changes now.
So again x squared goes into x to the 5,
x cubed times. So now, what are we going to get? We're going to get x to the 5
minus 2x to the 4,
and that's going to give me a 0
in Z 5, right, because again 3
minus negative 2, so 3 plus 2 is 5, and 5 is the same as 0
in Z 5.
So bring down the 0, that doesn't work, let's go down to the next term,
which is going to be 2x squared
plus 4x.
Now the question have to ask is how many times does…
so what do I have to multiply x squared by to get to 2, that's easy, it's going to be 2.
So that's going to give me 2x squared minus 4x.
4 minus [negative] 4 is 8, that's going to leave me with a remainder of 3x
plus 1.
And that is it. So here's my remainder, it’s now 3x plus 1,
and my quotient here is x cubed plus [2]. So if we go back to the previous example,
we have this longer polynomial, but if you reduce this mod 5,
we actually get the same answer. If you reduce the remainder mod 5,
if you look back, you get 3x plus 1.
I'm going to flip back and forth a little bit, so 48x plus 1,
48[x plus 1] mod 5 is 3x plus 1,
and over here our remainder is 3x plus 1. So,
in this setting, you do get the same answer if you just reduce the work
over R, over Z 5.
It's something to think about. Does it always happen?
I'll probably make another video where it may or may not happen.
I'm going to try a different one.
But that's the idea, okay?
So hopefully this gives you a little bit of practice [with] long division and thank you very much.