Hello everyone. In this video, I'd like to do one more quick example of long division.
So here we have 5x squared plus 1,
and I want to know what happens if I divide that into x cubed plus 3x squared plus 2x plus 1,
and I want to do this over R.
So my first step I need to
figure out how many times 5x squared goes into x cubed, that's 1 over
5x times.
So if I multiply those two things together I'm going to get x cubed
plus…maybe I'll write it over here.
So let’s…
let's write it over here, plus 1 over 5x,
just so that my subtraction
is easy to write.
That's going to give me 3x squared
and 2 minus a fifth
is 9 fifths
x plus 1.
Now lastly I want to know how many times 5x squared goes into 3x squared that's
plus 3 over 5 times.
So it’s going to be
3x squared,
and I'm going to get a plus 3 fifths.
Now I'm going to subtract,
and that's going to give me 9 over
5x
plus 2 over 5.
So again that's my remainder and up top is my quotient.
Now why did I do this example?
The reason why I did this example is for the following
interesting little tidbit here.
So this was the operation over R, now
in the last - so in a previous video I made,
the division worked out over R and if I just reduced the quotient and the remainder
over Z mod 5, everything worked out,
but here there's a huge problem,
and the huge problem is that well I'm dividing by 5 a lot, which I can't do
in Z mod 5 because 5 is 0
in Z mod 5.
So here you'd have to actually redo the computation.
It's not enough to do the computation over R and just reduce,
but here the computation over Z mod 5 is easy because
5x squared is the same as 0, so this is just dividing
the cubic polynomial by the polynomial given by 1,
and so there is no remainder, and the quotient
is just the original polynomial.
So something to keep in mind that you can't always just reduce the quotient and remainder.
You usually can - well I shouldn't say usually.
If you have integers and you do the computations over R,
there's some chance that you could reduce them, but it's not a given rule of thumb, okay?
You have to be a little bit careful about this fact. Okay,
that's all I want to say in this video. Hopefully
it helped clarify a couple of points that at least my class had
in class. Thank you very much.