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Current time:0:00Total duration:15:59

CCSS.Math:

In this video I want to
introduce you to the idea of a polynomial. It might sound like a really
fancy word, but really all it is is an expression that has a
bunch of variable or constant terms in them that are raised
to non-zero exponents. So that also probably
sounds complicated. So let me show you an example. If I were to give you x squared
plus 1, this is a polynomial. This is, in fact, a binomial
because it has two terms. The term polynomial is
more general. It's essentially saying you
have many terms. Poly tends to mean many. This is a binomial. If I were to say 4x to the third
minus 2 squared plus 7. This is a trinomial. I have three terms here. Let me give you just a more
concrete sense of what is and is not a polynomial. For example, if I were to have
x to the negative 1/2 plus 1, this is not a polynomial. That doesn't mean that you won't
ever see it while you're doing algebra or mathematics. But we just wouldn't call this
a polynomial because it has a negative and a fractional
exponent in it. Or if I were to give you the
expression y times the square root of y minus y squared. Once again, this is not a
polynomial, because it has a square root in it, which is
essentially raising something to the 1/2 power. So all of the exponents on our
variables are going to have to be non-negatives. Once again, neither of these
are polynomials. Now, when we're dealing with
polynomials, we're going to have some terminology. And you may or may not already
be familiar with it, so I'll expose it to you right now. The first terminology is the
degree of the polynomial. And essentially, that's the
highest exponent that we have in the polynomial. So for example, that polynomial
right there is a third degree polynomial. Now why is that? No need to keep writing it. Why is that a third
degree polynomial? Because the highest exponent
that we have in there is the x to the third term. So that's where we get it's
a third degree polynomial. This right here is a second
degree polynomial. And this is the second
degree term. Now a couple of other
terminologies, or words, that we need to know regarding
polynomials, are the constant versus the variable terms. And
I think you already know, these are variable
terms right here. This is a constant term. That right there is
a constant term. And then one last part to
dissect the polynomial properly is to understand the
coefficients of a polynomial. So let me write a fifth degree
polynomial here. And I'm going to write it in
maybe a non-conventional form right here. I'm going to not
do it in order. So let's just say it's x squared
minus 5x plus 7x to the fifth minus 5. So, once again, this is a
fifth degree polynomial. Why is that? Because the highest exponent
on a variable here is the 5 right here. So this tells us this is a
fifth degree polynomial. And you might say, well why do
we even care about that? And at least, in my mind, the
reason why I care about the degree of a polynomial is
because when the numbers get large, the highest degree term
is what really dominates all of the other terms. It will
grow the fastest, or go negative the fastest, depending
on whether there's a positive or a negative
in front of it. But it's going to dominate
everything else. It really gives you a sense for
how quickly, or how fast the whole expression would grow
or decrease in the case if it has a negative
coefficient. Now I just used the
word coefficient. What does that mean? Coefficient. And I've used it before,
when we were just doing linear equations. And coefficients are just the
constant terms that are multiplying the variable
terms. So for example, the coefficient
on this term right here is negative 5. You have to remember we have
a minus 5, so we consider negative 5 to be the
whole coefficient. The coefficient on
this term is a 7. There's no coefficient here;
it's just a constant term of negative 5. And then the coefficient on
the x squared term is 1. The coefficient is 1. It's implicit. You're assuming it's
1 times x squared. Now the last thing I want to
introduce you to is just the idea of the standard form
of a polynomial. Now none of this is going to
help you solve a polynomial just yet, but when we talk about
solving polynomials, I might use some of this
terminology, or your teacher might use some of this
terminology. So it's good to know what
we're talking about. The standard form of a
polynomial, essentially just list the terms in
order of degree. So this is in a non-standard
form. If I were to list this
polynomial in standard form, I would put this term first. So I
would write 7x to the fifth, then what's the next
smallest degree? Well, they have this
x squared term. I don't have an x to
the fourth or an x to the third here. So that'll be plus 1--
well I don't have to write 1-- plus x squared. And then I have this
term, minus 5x. And then I have this last term
right here, minus 5. This is the standard form of the
polynomial where you have it in descending order
of degree. Now let's do a couple of
operations with polynomials. And this is going to be a super
useful toolkit later on in your algebraic, or really in
your mathematical careers. So let's just simplify a
bunch of polynomials. And we've kind of touched on
this in previous videos. But I think this will give you
a better sense, especially when we have these higher
degree terms over here. So let's say I wanted to
add negative 2x squared plus 4x minus 12. And I'm going to add that
to 7x plus x squared. Now the important thing to
remember when you simplify these polynomials is that you're
going to add the terms of the same variable
of like degree. I'll do another example in a
second where I have multiple variables getting involved
in the situation. But anyway, I have these
parentheses here, but they really aren't doing anything. If I had a subtraction sign
here, I would have to distribute the subtraction,
but I don't. So I really could just write
this as minus 2x squared, plus 4x, minus 12, plus 7x,
plus x squared. And now let's simplify it. So let's add the terms
of like degree. And when I say like degree,
it has to also have the same variable. But in this example, we only
have the variable x. So let's add. Let's see, I have this x squared
term, and I've that x squared term, so I can
add them together. So I have minus 2x squared--
let me just write them together first --minus 2x
squared plus x squared. And then let me get the
x terms, so 4x and 7x. So this is plus 4x plus 7x. And then finally, I just
have this constant term right here, minus 12. And if I have negative 2 of
something, and I add 1 of something to that,
what do I have? Negative 2 plus 1 is negative
1x squared. I could just write negative
x squared. But I just want to show you that
I'm just adding negative 2 to 1 there. Then I have 4x plus 7x is 11x. And then I finally have my
constant term, minus 12. And I end up with a three
term, second degree polynomial. The leading coefficient here,
the coefficient on the highest degree term in standard form--
it's already in standard form --is negative 1. The coefficient here is 11. The constant term
is negative 12. Let's do another one
of these examples. I think you're getting
the general idea. Now let me do a complicated
example. So let's say I have 2a squared
b, minus 3ab squared, plus 5a squared b squared, minus 2a
squared b squared, plus 4a squared b, minus 5b squared. So here, I have a minus sign,
I have multiple variables. But let's just go through
this step by step. So the first thing
you want to do is distribute this minus sign. So this first part we can just
write as 2a squared b, minus 3ab squared, plus 5a
squared b squared. And then we want to distribute
this minus sign, or multiply all of these terms by negative
1 because we have the minus out here. So minus 2a squared b squared
minus 4a squared b. And a negative times a negative
is plus 5b squared. And now we want to essentially
add like terms. So I have this 2a squared b squared term. So do I have any other terms
that have an a squared b squared in them? Or sorry, a squared b. I have to be very
careful here. Well, that's ab squared, no.
a squared b squared. Oh! Here I have an a squared b. So let me write those
two down. So I have 2a squared b
minus 4a squared b. That's those two terms
right there. Let me go to orange. So here I have an
ab squared term. Now do I have any other
ab squared terms here? No, no other ab squared,
so I'll just write it: minus 3 ab squared. And then let's see, I have an a
squared b squared term here. Do I have any other ones? Well, yeah sure, the
next term is. That's an a squared b
squared term, so let me just write that. Plus 5a squared b
squared minus 2a squared b squared, right? I just wrote those two. And then finally I have that
last b squared term there, plus 5b squared. Now I can add them. So this first group right here
in this purplish color, 2 of something minus 4 of something
is going to be negative 2 of that something. So it's going to be negative
2a squared b. And then this term right here,
it's not going to add to anything, 3ab squared. And then we can add these two
terms. If I have 5 of something minus 2 of something,
I'm going to have 3 of that something. Plus 3a squared b squared. And then finally I have that
last term, plus 5b squared. We're done. We've simplified this
polynomial. Here, putting it in standard
form, you can think of it in different ways. The way I'd like to think of
it is maybe the combined degree of the term. Maybe we could put this one
first, but this is really according to your taste. So this is 3a squared
b squared. And then you could pick whether
you want to put the a squared b or the ab squared
terms first. 2a squared b. And then you have the
minus 3ab squared. And then we have just the
b squared term there. Plus 5b squared. And we're done. We've simplified this
polynomial. Now what I want to do next is
do a couple of examples of constructing a polynomial. And really, the idea is to give
you an appreciation for why polynomials are useful,
abstract representations. We're going to be using it all
the time, not only in algebra, but later in calculus, and
pretty much in everything. So they're really good things
to get familiar with. But what I want to do in these
four examples is represent the area of each of these figures
with a polynomial. And I'll try to match the colors
as closely as I can. So over here, what's the area? Well, this blue part right
here, the area there is x times y. And then what's the area here? It's going to be x times z. So plus x times z. But we have two of them! We have one x times z,
and then we have another x times z. So I could just add
an x times z here. Or I could just write, say,
plus 2 times x times z. And here we have a polynomial
that represents the area of this figure right there. Now let's do this next one. What's the area here? Well I have an a times a b. ab. This looks like an a times
a b again, plus ab. That looks like an ab
again, plus ab. I think they've drawn
it actually, a little bit strange. Well, I'm going to ignore
this c right there. Maybe they're telling us that
this right here is c. Because that's the information
we would need. Maybe they're telling us that
this base right there, that this right here, is c. Because that would help us. But if we assume that this is
another ab here, which I'll assume for this purpose
of this video. And then we have that last ab. And then we have this
one a times c. This is the area
of this figure. And obviously we can add
these four terms. This is 4ab and then
we have plus ac. And I made the assumption that
this was a bit of a typo, that that c where they were actually
telling us the width of this little square
over here. We don't know if it's a square,
that's only if a and c are the same. Now let's do this one. So how do we figure out the
area of the pink area? Well we could take the area
the whole rectangle, which would be 2xy, and then we could
subtract out the area of these squares. So each square has an area of
x times x, or x squared. And we have two of
these squares, so it's minus 2x squared. And then finally let's do
this one over here. So that looks like a dividing
line right there. So the area of this point, of
this area right there, is a times b, so it's ab. And then the area over here
looks like it will also be ab. So plus ab. And the area over
here is also ab. So the area here is 3ab. Anyway, hopefully that gets
us pretty warmed up with polynomials.