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Hello everyone. In this video, I'd like to do 
one more quick example of long division.

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So here we have 5x squared plus 1,

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and I want to know what happens if I divide that 
into x cubed plus 3x squared plus 2x plus 1,

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and I want to do this over R.

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So my first step I need to

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figure out how many times 5x squared 
goes into x cubed, that's 1 over

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5x times.

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So if I multiply those two things 
together I'm going to get x cubed

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plus…maybe I'll write it over here.

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So let’s…

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let's write it over here, plus 1 over 5x,

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just so that my subtraction

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is easy to write.

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That's going to give me 3x squared

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and 2 minus a fifth

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is 9 fifths

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x plus 1.

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Now lastly I want to know how many times 
5x squared goes into 3x squared that's

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plus 3 over 5 times.

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So it’s going to be

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3x squared,

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and I'm going to get a plus 3 fifths.

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Now I'm going to subtract,

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and that's going to give me 9 over

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5x

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plus 2 over 5.

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

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Now why did I do this example?

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The reason why I did this 
example is for the following

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interesting little tidbit here.

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So this was the operation over R, now

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in the last - so in a previous video I made,

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the division worked out over R and if I 
just reduced the quotient and the remainder 

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over Z mod 5, everything worked out,

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but here there's a huge problem,

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and the huge problem is that well I'm 
dividing by 5 a lot, which I can't do

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in Z mod 5 because 5 is 0

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in Z mod 5.

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So here you'd have to
actually redo the computation.

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It's not enough to do the computation 
over R and just reduce,

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but here the computation over
Z mod 5 is easy because

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5x squared is the same as 
0, so this is just dividing

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the cubic polynomial by the
polynomial given by 1,

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and so there is no remainder, and the quotient

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is just the original polynomial.

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So something to keep in mind that you can't 
always just reduce the quotient and remainder.

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You usually can - well I shouldn't say usually.

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If you have integers and you 
do the computations over R,

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there's some chance that you could reduce 
them, but it's not a given rule of thumb, okay?

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You have to be a little bit 
careful about this fact. Okay,

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that's all I want to say in this video. Hopefully

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it helped clarify a couple of 
points that at least my class had

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in class. Thank you very much.