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Hello everyone.

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In this video, we're going
to talk about long division,

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and we're going to do it over 
a couple of different fields. 

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So what are the quotient and remainder when

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this quintic polynomial is divided 
by this quadratic polynomial

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over the real numbers?

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And what if we do the
computations over Z 5, Z mod 5?

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What changes in this situation?

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Okay, so instead of typing this
out, I'm actually going to do this

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on paintbrush.

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So let's bring it up. Okay,

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so here I've written down the question again.

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We'd like to long divide 
these two polynomials.

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As you've noticed here in the middle,

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I have this extra x cubed
term with a 0 coefficient.

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It's just a placeholder to help me keep everything
organized as I attempt to go through this.

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Okay, so let's begin.

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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

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over R? So this computation is going to be 
over R, maybe I'll put a little R over here.

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Just to help remind us.

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Okay, so what number do I have to 
multiply x squared by to get to x 5,

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or what object, I guess, it's not really a number.

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x cubed.

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Now I take x cubed, multiply 
it by this whole polynomial,

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what do I get? I'm going to get x to the 5.

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If these don't match up, then
you've made a mistake up here,

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and I'm going to get minus 2x

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to the 3 plus 1 which is 4.

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Now I subtract,

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and that's going to give me 5x to the 4.

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Now I'm going to bring down the 0 term,

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again it's not important that you 
write this. I'm just doing it for

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organizational purposes. 

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x squared, now the number I'm going to 
multiply to get to 5x to the 4 that's 5x squared.

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So that's going to give me

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5x to the 4,

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and then I'm going to take 5x 
squared multiply by the minus 2x,

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that’s going to be minus 10x cubed. 

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Now we subtract.

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First two terms cancel again. Next two 
terms, it's going to give me a 10x cubed. 

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Okay?

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Plus, let's bring down the next term, 2x squared.

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x squared multiplied by 10x is 10x cubed,

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so up top I go 10x…

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by the way, another way to 
see this, doesn't always work -

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well it does always work we have to 
think of the word divide carefully,

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you could divide 10x cubed by x squared,

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and that will give you the number up there.

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You have to divide properly in 
whatever field you're working in.

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In this case, it's R, so we have a good 
understanding of what's going on.

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10x cubed

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and minus 20x squared,

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so it should be a 2.

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Subtract, 

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so it's going to be 22

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x squared.

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I'm going to drop down the 4x,

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I'm going to do the long 
division so it goes in 22 times.

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Okay? So I'm going to get

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22x squared

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minus, what is that, 44x.

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And my last subtraction

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is going to give me 48x

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plus 1.

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And remember that this is the remainder,

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the 48x plus 1, so maybe I'll
write the word remainder.

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So that's my remainder, 
and up top is my quotient.

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Okay, great. So there's a long 
division example over R, okay?

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Now what I'd like to do is I'd like to do 
the same computation over Z mod 5

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and see what changes, okay?

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A lot of - okay, and we'll see what changes.

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Over Z mod 5.

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Okay, so let's see what changes now.

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So again x squared goes into x to the 5,

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x cubed times. So now, what are we going 
to get? We're going to get x to the 5 

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minus 2x to the 4,

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and that's going to give me a 0

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in Z 5, right, because again 3

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minus negative 2, so 3 plus 2 
is 5, and 5 is the same as 0

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in Z 5.

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So bring down the 0, that doesn't 
work, let's go down to the next term,

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which is going to be 2x squared

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plus 4x.

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Now the question have to ask 
is how many times does…

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so what do I have to multiply x squared by
to get to 2, that's easy, it's going to be 2.

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So that's going to give 
me 2x squared minus 4x.

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4 minus [negative] 4 is 8, that's going 
to leave me with a remainder of 3x

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plus 1.

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And that is it. So here's my 
remainder, it’s now 3x plus 1, 

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and my quotient here is x cubed plus [2]. 
So if we go back to the previous example,

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we have this longer polynomial, 
but if you reduce this mod 5,

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we actually get the same answer. If 
you reduce the remainder mod 5,

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if you look back, you get 3x plus 1.

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I'm going to flip back and 
forth a little bit, so 48x plus 1,

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48[x plus 1] mod 5 is 3x plus 1,

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and over here our
remainder is 3x plus 1. So,

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in this setting, you do get the same 
answer if you just reduce the work

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over R, over Z 5.

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It's something to think about.
Does it always happen?

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I'll probably make another video 
where it may or may not happen. 

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I'm going to try a different one.

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But that's the idea, okay?

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So hopefully this gives you a little bit of practice 
[with] long division and thank you very much.