Now I want to talk about the polynomial ring, okay, so we're going to finish this week off
with a little bit of an introduction to polynomials.
So definition, a polynomial in x over a ring R
is an expression of the form:
a n times x to the n plus dot dot dot plus a1 times x plus a0,
where all of these elements are actually in your ring R,
and n greater than or equal to 0 is an integer.
We're going to [denote] the set, or the ring, of all polynomials
over R by R of x, okay?
I say this is a ring that means I'm going to have to give it an addition and multiplication structure.
It's what you think it is, it's what you've been doing all along with polynomials.
You've been multiplying polynomials for a long time.
Do that operation for multiplication, and addition is, again, what you think, just
common factor like-terms with the powers of x’s.
As a reminder, what's a ring?
Remember that a ring is a set with two operations,
addition and multiplication, such that it satisfies some usual axioms, okay? That you can…
there's additive inverses, you can multiply,
there's the distributive property, it's commutative and associative,
there's multiplicative and additive identities, things like this.
Typically in this course,
we're going to use the definition of a polynomial over a field. Okay?
This makes our life easier, but I do want to mention
that you can define a polynomial ring over a ring.
You can even define it more generally, but I think it makes…
it’s best to do it over a ring like we've done here.
What else is important here to mention?
Alright, so I mentioned that…
yeah we will do this over a field. Our fields will include the rational numbers, the real numbers,
the complex numbers, and Z mod p.
Those are the ones that we use, but you could also talk about
let's say the integral polynomials, right?
Polynomials with integer coefficients and things like that.
I just don’t want our definition to restrict us to a field when we know what
the word ring means and we can talk about it a little bit more general.
So assorted definitions, so a lot of these you already know.
I'm going to just do some highlights from this.
The degree of a polynomial is the largest non-zero term.
So we're assuming here that a n is not 0,
so that gives us the degree of the polynomial.
The largest non-zero…
the largest non-zero term.
Whatever the power of x is for that term, we call that n.
The degree of the 0 polynomial is undefined, we sometimes call it negative infinity,
we'll see why in a little bit,
but I think for us we're just going to say it's undefined, that’s perfectly fine for us.
Two polynomials are equal if and only if their degrees are the same and
each of the coefficients match up, term by term.
x is an indeterminate, so something to keep in mind here,
this x thing doesn't have any meaning, okay?
One thing we can do is we can attach meaning to x,
so for example if we look at a complex polynomial
and we'd like to evaluate it at a value, that's fine, but that's
doing something a little bit more general than just like -
so plugging in a number is actually defining a function is kind of what's happening here,
the function takes the polynomial and plugs in, instead of x, some number
and then simplifies it. That's really what's happening there.
So you can view this as a map, it's a little bit abstract but I had a couple of students
really confused - well not really confused but
wondering about, “Okay what is this x thing?
Like you keep changing it sometimes, and sometimes you don’t,”
and that's really what's happening, okay? So the polynomial...
the polynomial has
coefficients over, let's say, a ring or a field,
and the x has no meaning, it's just an indeterminate, okay?
You can give it meaning, you can plug in values and substitute and
and things like that, but that's defining a map that goes into
a different ring, etc, etc, okay?
It's a subtle point, so if you miss that it’s not a huge deal, but
if you really understand this and you’re wondering, “Oh why can’t I do these things?” This is what's
happening, hopefully it gives you a little bit of an idea.
Operations on polynomials: addition, subtraction, multiplication.
They're what you expect they are. Addition, again,
common factor like-terms, multiplication, just FOIL it out like you normally would.
Last thing I want to talk about is the Division Algorithm for Polynomials. So
there's not too much - this is a…the introduction to polynomial says a lot of words and
just making sure that we're on the same page of definitions and things like that.
One thing that we did do though that was a little bit of substance
is the Division Algorithm for Polynomials.
So here let F be a field, so remember for us if you want to simplify your life a lot,
think of a field as Q, R, C, or Z mod p.
If f and g are polynomials in our field, and g of x is not 0,
and there exist unique polynomials q of x and r of x inside
f of x such that
f of x is equal to q x times g x plus r of x,
with r x equals 0, or the degree of r is less than the degree of g.
So the Division Algorithm for Polynomials, this is why sometimes people say that
the degree of the 0 polynomial is negative infinity.
It works in theorems like this for example, right, where
negative infinity would be less than the degree of g
for any g as long as g was non-zero
But we know that g is non-zero, we have that right there.
Good, good, good.
So this is a polynomial, notice that this is very similar to
Division Algorithm for Integers. If we have a is equal to
b q plus r, instead now we have f is equal to q g plus r, it's the same idea.
And it's the same idea, it's kind of like evaluating an f over g,
but we're doing it as f is equal to q g plus r of x.
One thing we're going to talk about, maybe I’ll start the next one with this, is
Long Division of Polynomials, that's something that's pretty important.
Long division for us is going to be a little bit more complicated than normal because
our fields might have really weird coefficients, right? We might be working…
divide two polynomials over Z 7, let's say.
That's a little different, right, than doing it over R.
There's a lots of examples of this, I do have a couple of videos where I do do the long division
process. You can check them out on the website,
but I think that's all I have to say.
So thank you very much your time, thanks for listening,
and hopefully this video gives you
a little bit of insight into Week 10 of Math 135. Take care.