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Now I want to talk about the polynomial ring, 
okay, so we're going to finish this week off

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with a little bit of an introduction to polynomials.

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So definition, a polynomial 
in x over a ring R

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is an expression of the form:

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a n times x to the n plus dot dot 
dot plus a1 times x plus a0, 

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where all of these elements 
are actually in your ring R,

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and n greater than or 
equal to 0 is an integer.

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We're going to [denote] the set, 
or the ring, of all polynomials

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over R by R of x, okay?

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I say this is a ring that means I'm going to have 
to give it an addition and multiplication structure.

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It's what you think it is, it's what you've 
been doing all along with polynomials.

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You've been multiplying 
polynomials for a long time.

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Do that operation for multiplication, and
addition is, again, what you think, just

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common factor like-terms
with the powers of x’s.

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As a reminder, what's a ring?

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Remember that a ring is
a set with two operations,

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addition and multiplication, such that it satisfies
some usual axioms, okay? That you can…

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there's additive inverses, you can multiply,

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there's the distributive property, 
it's commutative and associative,

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there's multiplicative and
additive identities, things like this.

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Typically in this course,

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we're going to use the definition of 
a polynomial over a field. Okay?

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This makes our life easier,
but I do want to mention

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that you can define a 
polynomial ring over a ring.

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You can even define it more 
generally, but I think it makes…

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it’s best to do it over a
ring like we've done here.

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What else is important here to mention?

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Alright, so I mentioned that…

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yeah we will do this over a field. Our fields will 
include the rational numbers, the real numbers,

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the complex numbers, and Z mod p.

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Those are the ones that we use, 
but you could also talk about

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let's say the integral
polynomials, right?

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Polynomials with integer 
coefficients and things like that.

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I just don’t want our definition to
restrict us to a field when we know what

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the word ring means and we can 
talk about it a little bit more general. 

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So assorted definitions, so a 
lot of these you already know.

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I'm going to just do 
some highlights from this.

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The degree of a polynomial 
is the largest non-zero term.

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So we're assuming here that a n is not 0,

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so that gives us the degree of the polynomial.

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The largest non-zero…

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the largest non-zero term.

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Whatever the power of x is 
for that term, we call that n.

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The degree of the 0 polynomial is undefined,
we sometimes call it negative infinity,

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we'll see why in a little bit,

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but I think for us we're just going to say 
it's undefined, that’s perfectly fine for us.

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Two polynomials are equal if and only 
if their degrees are the same and

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each of the coefficients 
match up, term by term.

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x is an indeterminate, so 
something to keep in mind here,

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this x thing doesn't have 
any meaning, okay?

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One thing we can do is we 
can attach meaning to x,

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so for example if we look 
at a complex polynomial

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and we'd like to evaluate it at 
a value, that's fine, but that's

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doing something a little bit 
more general than just like - 

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so plugging in a number is actually defining 
a function is kind of what's happening here,

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the function takes the polynomial and 
plugs in, instead of x, some number

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and then simplifies it. That's 
really what's happening there.

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So you can view this as a map, it's a little 
bit abstract but I had a couple of students

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really confused - well not really confused but

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wondering about, “Okay what is this x thing?

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Like you keep changing it sometimes, 
and sometimes you don’t,” 

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and that's really what's happening, 
okay? So the polynomial...

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the polynomial has

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coefficients over, let's say, a ring or a field,

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and the x has no meaning, it's 
just an indeterminate, okay?

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You can give it meaning, you can 
plug in values and substitute and

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and things like that, but that's 
defining a map that goes into

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a different ring, etc, etc, okay?

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It's a subtle point, so if you miss 
that it’s not a huge deal, but

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if you really understand this and you’re wondering,
“Oh why can’t I do these things?” This is what's

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happening, hopefully it gives
you a little bit of an idea.

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Operations on polynomials: 
addition, subtraction, multiplication. 

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They're what you expect 
they are. Addition, again,

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common factor like-terms, multiplication, 
just FOIL it out like you normally would.

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Last thing I want to talk about is the
Division Algorithm for Polynomials. So

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there's not too much - this is a…the introduction 
to polynomial says a lot of words and

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just making sure that we're on the same 
page of definitions and things like that.

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One thing that we did do though 
that was a little bit of substance

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is the Division Algorithm for Polynomials.

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So here let F be a field, so remember for 
us if you want to simplify your life a lot,

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think of a field as Q, R, C, or Z mod p.

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If f and g are polynomials in 
our field, and g of x is not 0,

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and there exist unique polynomials 
q of x and r of x inside

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f of x such that

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f of x is equal to q x 
times g x plus r of x,

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with r x equals 0, or the degree 
of r is less than the degree of g.

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So the Division Algorithm for Polynomials, 
this is why sometimes people say that

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the degree of the 0 polynomial is negative infinity.

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It works in theorems like this 
for example, right, where

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negative infinity would be 
less than the degree of g

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for any g as long
as g was non-zero

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But we know that g is non-zero, 
we have that right there.

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Good, good, good.

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So this is a polynomial, notice 
that this is very similar to

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Division Algorithm for Integers. 
If we have a is equal to

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b q plus r, instead now we have f is 
equal to q g plus r, it's the same idea.

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And it's the same idea, it's kind 
of like evaluating an f over g,

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but we're doing it as f 
is equal to q g plus r of x.

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One thing we're going to talk about, 
maybe I’ll start the next one with this, is

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Long Division of Polynomials, that's 
something that's pretty important.

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Long division for us is going to be a little 
bit more complicated than normal because

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our fields might have really weird
coefficients, right? We might be working…

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divide two polynomials over Z 7, let's say.

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That's a little different, 
right, than doing it over R.

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There's a lots of examples of this, I do have a 
couple of videos where I do do the long division

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process. You can check
them out on the website,

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but I think that's all I have to say.

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So thank you very much your 
time, thanks for listening,

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and hopefully this video gives you

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a little bit of insight into Week
10 of Math 135. Take care.