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In this video I want to introduce you to the idea of a

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polynomial.

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It might sound like a really fancy word, but really all it

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is is an expression that has a bunch of variable or constant

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terms in them that are raised to non-zero exponents.

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So that also probably sounds complicated.

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So let me show you an example.

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If I were to give you x squared plus 1, this is a

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polynomial.

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This is, in fact, a binomial because it has two terms. The

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term polynomial is more general.

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It's essentially saying you have many terms. Poly

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tends to mean many.

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This is a binomial.

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If I were to say 4x to the third minus 2 squared plus 7.

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This is a trinomial.

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I have three terms here.

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Let me give you just a more concrete sense of what is and

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is not a polynomial.

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For example, if I were to have x to the negative 1/2 plus 1,

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this is not a polynomial.

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That doesn't mean that you won't ever see it while you're

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doing algebra or mathematics.

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But we just wouldn't call this a polynomial because it has a

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negative and a fractional exponent in it.

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Or if I were to give you the expression y times the square

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root of y minus y squared.

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Once again, this is not a polynomial, because it has a

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square root in it, which is essentially raising something

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to the 1/2 power.

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So all of the exponents on our variables are going to have to

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be non-negatives.

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Once again, neither of these are polynomials.

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Now, when we're dealing with polynomials, we're going to

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have some terminology.

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And you may or may not already be familiar with it, so I'll

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expose it to you right now.

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The first terminology is the degree of the polynomial.

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And essentially, that's the highest exponent that we have

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in the polynomial.

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So for example, that polynomial right there is a

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third degree polynomial.

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Now why is that?

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No need to keep writing it.

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Why is that a third degree polynomial?

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Because the highest exponent that we have in there is the x

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to the third term.

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So that's where we get it's a third degree polynomial.

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This right here is a second degree polynomial.

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And this is the second degree term.

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Now a couple of other terminologies, or words, that

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we need to know regarding polynomials, are the constant

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versus the variable terms. And I think you already know,

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these are variable terms right here.

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This is a constant term.

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That right there is a constant term.

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And then one last part to dissect the polynomial

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properly is to understand the coefficients of a polynomial.

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So let me write a fifth degree polynomial here.

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And I'm going to write it in maybe a non-conventional form

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right here.

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I'm going to not do it in order.

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So let's just say it's x squared minus 5x plus 7x to

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the fifth minus 5.

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So, once again, this is a fifth degree polynomial.

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Why is that?

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Because the highest exponent on a variable here is the 5

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right here.

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So this tells us this is a fifth degree polynomial.

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And you might say, well why do we even care about that?

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And at least, in my mind, the reason why I care about the

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degree of a polynomial is because when the numbers get

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large, the highest degree term is what really dominates all

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of the other terms. It will grow the fastest, or go

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negative the fastest, depending on whether there's a

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positive or a negative in front of it.

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But it's going to dominate everything else.

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It really gives you a sense for how quickly, or how fast

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the whole expression would grow or decrease in the case

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if it has a negative coefficient.

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Now I just used the word coefficient.

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What does that mean?

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Coefficient.

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And I've used it before, when we were just

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doing linear equations.

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And coefficients are just the constant terms that are

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multiplying the variable terms.

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So for example, the coefficient on this term right

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here is negative 5.

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You have to remember we have a minus 5, so we consider

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negative 5 to be the whole coefficient.

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The coefficient on this term is a 7.

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There's no coefficient here; it's just a constant term of

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negative 5.

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And then the coefficient on the x squared term is 1.

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The coefficient is 1.

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It's implicit.

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You're assuming it's 1 times x squared.

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Now the last thing I want to introduce you to is just the

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idea of the standard form of a polynomial.

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Now none of this is going to help you solve a polynomial

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just yet, but when we talk about solving polynomials, I

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might use some of this terminology, or your teacher

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might use some of this terminology.

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So it's good to know what we're talking about.

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The standard form of a polynomial, essentially just

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list the terms in order of degree.

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So this is in a non-standard form.

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If I were to list this polynomial in standard form, I

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would put this term first. So I would write 7x to the fifth,

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then what's the next smallest degree?

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Well, they have this x squared term.

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I don't have an x to the fourth or an x

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to the third here.

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So that'll be plus 1-- well I don't have to

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write 1-- plus x squared.

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And then I have this term, minus 5x.

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And then I have this last term right here, minus 5.

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This is the standard form of the polynomial where you have

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it in descending order of degree.

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Now let's do a couple of operations with polynomials.

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And this is going to be a super useful toolkit later on

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in your algebraic, or really in your mathematical careers.

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So let's just simplify a bunch of polynomials.

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And we've kind of touched on this in previous videos.

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But I think this will give you a better sense, especially

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when we have these higher degree terms over here.

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So let's say I wanted to add negative 2x squared

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plus 4x minus 12.

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And I'm going to add that to 7x plus x squared.

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Now the important thing to remember when you simplify

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these polynomials is that you're going to add the terms

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of the same variable of like degree.

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I'll do another example in a second where I have multiple

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variables getting involved in the situation.

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But anyway, I have these parentheses here, but they

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really aren't doing anything.

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If I had a subtraction sign here, I would have to

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distribute the subtraction, but I don't.

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So I really could just write this as minus 2x squared, plus

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4x, minus 12, plus 7x, plus x squared.

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And now let's simplify it.

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So let's add the terms of like degree.

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And when I say like degree, it has to also

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have the same variable.

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But in this example, we only have the variable x.

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So let's add.

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Let's see, I have this x squared term, and I've that x

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squared term, so I can add them together.

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So I have minus 2x squared-- let me just write them

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together first --minus 2x squared plus x squared.

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And then let me get the x terms, so 4x and 7x.

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So this is plus 4x plus 7x.

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And then finally, I just have this constant term

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right here, minus 12.

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And if I have negative 2 of something, and I add 1 of

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something to that, what do I have?

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Negative 2 plus 1 is negative 1x squared.

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I could just write negative x squared.

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But I just want to show you that I'm just adding negative

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2 to 1 there.

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Then I have 4x plus 7x is 11x.

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And then I finally have my constant term, minus 12.

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And I end up with a three term, second degree

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polynomial.

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The leading coefficient here, the coefficient on the highest

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degree term in standard form-- it's already in standard form

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--is negative 1.

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The coefficient here is 11.

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The constant term is negative 12.

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Let's do another one of these examples.

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I think you're getting the general idea.

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Now let me do a complicated example.

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So let's say I have 2a squared b, minus 3ab squared, plus 5a

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squared b squared, minus 2a squared b squared, plus 4a

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squared b, minus 5b squared.

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So here, I have a minus sign, I have multiple variables.

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But let's just go through this step by step.

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So the first thing you want to do is

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distribute this minus sign.

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So this first part we can just write as 2a squared b, minus

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3ab squared, plus 5a squared b squared.

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And then we want to distribute this minus sign, or multiply

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all of these terms by negative 1 because we have

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the minus out here.

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So minus 2a squared b squared minus 4a squared b.

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And a negative times a negative is plus 5b squared.

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And now we want to essentially add like terms. So I have this

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2a squared b squared term.

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So do I have any other terms that have an a squared b

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squared in them?

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Or sorry, a squared b.

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I have to be very careful here.

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Well, that's ab squared, no. a squared b squared.

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Oh!

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Here I have an a squared b.

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So let me write those two down.

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So I have 2a squared b minus 4a squared b.

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That's those two terms right there.

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Let me go to orange.

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So here I have an ab squared term.

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Now do I have any other ab squared terms here?

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No, no other ab squared, so I'll just write

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it: minus 3 ab squared.

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And then let's see, I have an a squared b squared term here.

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Do I have any other ones?

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Well, yeah sure, the next term is.

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That's an a squared b squared term, so let

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me just write that.

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Plus 5a squared b squared minus 2a

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squared b squared, right?

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I just wrote those two.

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And then finally I have that last b squared term there,

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plus 5b squared.

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Now I can add them.

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So this first group right here in this purplish color, 2 of

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something minus 4 of something is going to be negative 2 of

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that something.

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So it's going to be negative 2a squared b.

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And then this term right here, it's not going to add to

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anything, 3ab squared.

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And then we can add these two terms. If I have 5 of

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something minus 2 of something, I'm going to have 3

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of that something.

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Plus 3a squared b squared.

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And then finally I have that last term, plus 5b squared.

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We're done.

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We've simplified this polynomial.

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Here, putting it in standard form, you can think of it in

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different ways.

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The way I'd like to think of it is maybe the combined

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degree of the term.

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Maybe we could put this one first, but this is really

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according to your taste.

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So this is 3a squared b squared.

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And then you could pick whether you want to put the a

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squared b or the ab squared terms first. 2a squared b.

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And then you have the minus 3ab squared.

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And then we have just the b squared term there.

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Plus 5b squared.

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And we're done.

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We've simplified this polynomial.

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Now what I want to do next is do a couple of examples of

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constructing a polynomial.

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And really, the idea is to give you an appreciation for

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why polynomials are useful, abstract representations.

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We're going to be using it all the time, not only in algebra,

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but later in calculus, and pretty much in everything.

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So they're really good things to get familiar with.

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But what I want to do in these four examples is represent the

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area of each of these figures with a polynomial.

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And I'll try to match the colors as closely as I can.

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So over here, what's the area?

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Well, this blue part right here, the area

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there is x times y.

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And then what's the area here?

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It's going to be x times z.

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So plus x times z.

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But we have two of them!

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We have one x times z, and then we have

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another x times z.

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So I could just add an x times z here.

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Or I could just write, say, plus 2 times x times z.

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And here we have a polynomial that represents the area of

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this figure right there.

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Now let's do this next one.

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What's the area here?

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Well I have an a times a b.

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ab.

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This looks like an a times a b again, plus ab.

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That looks like an ab again, plus ab.

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I think they've drawn it actually,

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a little bit strange.

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Well, I'm going to ignore this c right there.

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Maybe they're telling us that this right here is c.

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Because that's the information we would need.

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Maybe they're telling us that this base right there, that

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this right here, is c.

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00:14:21,240 --> 00:14:22,850
Because that would help us.

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But if we assume that this is another ab here, which I'll

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assume for this purpose of this video.

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And then we have that last ab.

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And then we have this one a times c.

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This is the area of this figure.

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00:14:39,549 --> 00:14:41,579
And obviously we can add these four terms.

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This is 4ab and then we have plus ac.

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00:14:46,779 --> 00:14:49,399
And I made the assumption that this was a bit of a typo, that

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that c where they were actually telling us the width

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00:14:52,309 --> 00:14:54,239
of this little square over here.

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We don't know if it's a square, that's only if a and c

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are the same.

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Now let's do this one.

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So how do we figure out the area of the pink area?

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00:15:03,190 --> 00:15:06,010
Well we could take the area the whole rectangle, which

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00:15:06,009 --> 00:15:10,730
would be 2xy, and then we could subtract out the area of

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00:15:10,730 --> 00:15:12,240
these squares.

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00:15:12,240 --> 00:15:18,060
So each square has an area of x times x, or x squared.

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00:15:18,059 --> 00:15:19,809
And we have two of these squares, so

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it's minus 2x squared.

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00:15:23,190 --> 00:15:27,050
And then finally let's do this one over here.

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00:15:27,049 --> 00:15:30,429
So that looks like a dividing line right there.

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00:15:30,429 --> 00:15:34,250
So the area of this point, of this area right there, is a

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00:15:34,250 --> 00:15:37,129
times b, so it's ab.

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00:15:37,129 --> 00:15:43,679
And then the area over here looks like it will also be ab.

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00:15:43,679 --> 00:15:45,319
So plus ab.

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00:15:45,320 --> 00:15:50,950
And the area over here is also ab.

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00:15:50,950 --> 00:15:54,170
So the area here is 3ab.

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00:15:54,169 --> 00:15:57,699
Anyway, hopefully that gets us pretty warmed up with

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00:15:57,700 --> 00:15:58,950
polynomials.

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