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- 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.