WEBVTT mathematics/math-analysis/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to have an introduction to polynomials.
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By this point, you have seen polynomials, even if you don't remember the name, countless times in previous courses.
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As a brief reminder, they are the ones that look like x² - 2x + 9,
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or maybe 3x⁵ - 8x³ + 10x² + x + 47.
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This stuff looks familiar; now, you might wonder why you have spent so much time on them before,
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and why we are studying them yet again in another course.
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In short, it is because polynomials are ridiculously, absurdly useful.
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They come up in every branch of science, from physics to medicine to economics.
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They are going to be important if you are going to do engineering work; they are going to be important if you are going to do computer programming work.
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They are going to be important for pretty much anything you want to do.
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If you want to study higher-level mathematics, they are going to be important in that, too.
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Polynomials are very important; they are going to be important in any branch of science, and in anything that is in higher, deeper levels of mathematics.
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So, that is why they keep drilling them for all these years--because you really have to understand polynomials
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for a huge number of things, so it is really important to get a good grasp on it now.
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A polynomial: what is a polynomial? Formally, we define it as an expression of the form
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a<font size="-6">n</font>x^n + a<font size="-6">n - 1</font>x^n - 1 +...+ a₂ times x² + a₁x + a₀.
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And now, don't worry; these little things down here we just call the subscripts, which just means to say that there is a,
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but then there are many different a's; there is a<font size="-6">n</font>, a<font size="-6">n - 1</font>, a<font size="-6">n - 2</font>, and so on and so on...
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a₂, a₁, a₀...just many different a's.
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What this expression means: we have that n is a non-negative integer, and all of our a's
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(the a<font size="-6">n</font>, a<font size="-6">n - 1</font>, and so on, up until a₀), are all real numbers,
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which is to say that they are just constants; and finally, a<font size="-6">n</font> itself, the first one,
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the one at the very front, is not equal to 0.
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Now, that might seem a little complex in its formal definition; but don't worry;
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we are about to explain what is going on, so we can really understand what a polynomial is.
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So, our expression, once again, was a<font size="-6">n</font>x^n + a<font size="-6">n - 1</font>x^n - 1,
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and so on and so on...a₁x + a₀.
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The first thing that we want to get to is: we want to start with this non-negative integer, n.
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This n is really important; that n can be any number...something like 1 or 5 or 968.
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It is just the exponent that the very first x has; so we could have x¹
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(which we would normally write as just x), and then other stuff after it.
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Or we could have x⁵, and then other stuff after it; or we could have x^968, and then other stuff after it.
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The n is basically our starting point--what is our starting exponent going to be?
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Then, we have this structure: _x^n + _x^n - 1 + _x^n - 2...so on and so on,
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until finally we get _x² + _x + _.
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If we took x⁵, then we would have _x⁵ + _x⁴ + _x³ + _x² + _x + _.
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We just fill in those blanks with numbers.
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That is what all of these a's represent; these are our blanks, down below.
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They are the things that we are filling in; the a's represent those blanks.
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They are just a number that is going to get stuffed into that place.
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And finally, the a's (blank spaces) above can be, potentially, zeroes; so if we had a 0 here, we would just knock out the whole thing.
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And we would pretend it wasn't there; we would read it as x^n +...and then x^n - 2 would be next.
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If we have _x² + _x + _, and we have 5x² + 0x + 3,
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we would probably just read this as 5x² + 3.
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So, if we have an a as a 0, it can cause that spot to just disappear.
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Now, the only spot that is not allowed to disappear is a<font size="-6">n</font>: a<font size="-6">n</font>, the first spot, this one up here, is not allowed to be 0.
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Why not? Because, if it was 0, then our x^n would just disappear.
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If we were able to have 0, then it would be gone; and so, if it is gone, our x^n would disappear,
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at which point, why did we choose n in the first place, if we are not even going to have x^n show up?
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So, since we want to use n (that is why we chose n), we can't have our very first spot disappear and get rid of that n.
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And that is it--that is a polynomial: _x to the exponent, plus _x to the other exponent, plus blank...and so on and so forth.
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That is pretty much just the structure of a polynomial.
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If you can remember that, that is the important part.
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While a polynomial is technically just an expression, like, for example, x⁴ + 3x² - 9x + 17--
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a polynomial is just this expression of _x to the exponent + _x to the exponent + _x to the exponent--
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that is all it is--just that structure of _x to the exponent--we normally use them to make functions or equations.
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So, a polynomial function is just a function that has been made out of a polynomial.
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A polynomial function is a function that is equal to some polynomial.
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And a polynomial equation is just an equation made out of it, as well.
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So, we could have y = polynomial, or we could have function = polynomial.
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That is it; also, while we will generally use x as the variable in polynomials, we should note that any variable can be used.
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Any variable can be used; the important thing is that we are just following this _something to the exponent structure.
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Like in our work with functions, we normally use f(x); but there is no reason that we have to use x.
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x is a commonly-used variable, but it is not the only one out there.
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There are others out there; so all of the below are just as valid as x⁴ + 3x² - 9x + 17.
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We could have z⁴ + 3z² - 9z + 17--
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representing the same thing; but instead, now we have a different variable being the placeholder.
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Or we could have l to the fourth and more things, or θ to the fourth.
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Any symbol can be our placeholder; we just want something that is being that placeholder, and being raised to an exponent.
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The degree of the polynomial is the value of n in this expression; it is whatever our highest exponent is at the front.
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Informally, we just want to see it as...the degree of the largest exponent on a variable.
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So, that is what we want to think of degree as: the largest exponent on one of our variables.
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If the polynomial isn't in order of largest to smallest exponents...
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Normally we are in order--we go to n, and then next we are at n - 1, and then next we would be at x^n - 2, and so on
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and so on and so on, until eventually we got to x², and then x¹, and then...
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although you might not remember this from exponent work before, x⁰, which we will talk about
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in exponents later on--but the point is that we keep lowering the exponent--
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we keep going and going and going, until we are finally at a constant.
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But if our polynomial isn't in order of largest to smallest exponents, the degree might not necessarily be the very first one that you see.
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It might not necessarily be the one at the very beginning; it could be somewhere in the middle,
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if we aren't necessarily in that order of largest to smallest exponents.
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The important thing is just to find the largest exponent on a variable; and that is your degree.
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Let's see some examples: we could have a polynomial x² + 2x + 1.
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We look at this one, and we say, "Oh, the largest exponent on anything is that 2"; so we get a degree of 2.
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We look at this one, 5x + 3; and the biggest one here is just this x.
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What is its exponent? The exponent of anything is just to the 1, if it doesn't have something already, so we get 1.
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We look at the next one: 7x³ - 4x^47 + 8.
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The one at the front is x³, but it isn't going to be our degree.
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The degree ends up being...this one isn't in our usual order; it isn't in that general form
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of x^n and then x^n - 1 and then x^n - 2; this one is out of order.
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But that doesn't mean that we can't find its degree; we just look through.
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We look at all of our x's, and we end up seeing that 47 is the largest exponent on any of our variables.
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And so, it is 47 that is our degree.
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Finally, the last one might be a little bit confusing, as well.
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We see this one, and we think, "Oh, x³...wait, there is an even larger exponent here."
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We have 3⁵, but 3 is not an x; it is not a variable.
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So, since it is not a variable, it is out of the running, which leaves us with x³ as what we have.
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And so, the degree of that is 3; so you are looking for a variable (make sure it is a variable) with the highest, largest exponent.
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And that is your degree for a polynomial.
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Since a lot of different polynomials come up very often, we have some special names for them.
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Some types of polynomials get special names, and so we want to know them.
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They are not super important to remember, although quadratic will come up so often, it is definitely going to be burned into your memory.
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It is not super important to absolutely remember these; but they will come up.
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And so, you want to know them, because you might have to know these vocabulary words.
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You can figure out what name to use, based on the degree of a polynomial, for these ones.
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A cubic is a degree 3 polynomial; this one has a degree of 3 here; or 5x³ - 3x² + 27, once again, has a degree of 3.
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A quadratic has a degree of 2; so it is x² + x + 1 or -17x² + 20x - √2.
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A linear has a degree of 1: x¹, πx¹...
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And then finally, a constant is just a degree 0 polynomial, which is to say it has no variables in it at all.
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So, 1 has no variables; 5,111,723 still has no variable--there is no x here, so since there is no x, we have degree 0.
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We can also talk about a polynomial based on the number of terms that make it up.
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Once again, it is not super important to have this really memorized; but you want to be familiar with
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and aware of these vocabulary terms, because they will show up now and then.
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A trinomial is something that has three terms; we can remember this from *tri*nomial,
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like a tricycle or a triangle--they are all things having to do with the number 3.
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x² + x + 1: the squared isn't so much the important part as the x²; we have three things.
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47x⁹ + x³ + 2: the degree no longer matters.
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It is not about the degree, so I really should not have accidentally circled that 2...x²...
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It is just the number of things we have: 47x⁹ + x³ + 2...
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A binomial is something that has two terms; x and then 1, or -52x⁷ and 892x.
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It doesn't matter that it is a coefficient times an x; that is OK.
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It is allowed to be a coefficient times some variable raised to some exponent.
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But that is the whole thing--that is one of our terms for this.
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A binomial has two terms; it could be x + 1 (as simple as that), or it could be more complex, like -52x⁷ + 892x.
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Or we could have one term, which is x, or maybe even something really, really large, like x raised to the 1,845.
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All right, the distributive property: very often, we are going to need to either factor polynomials--
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break them into their multiplicative pieces--or expand these factors into a polynomial that is in general form.
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So, take these multiplicative pieces, and then combine them together to get something larger
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that gives us the whole polynomial in that general form that we saw of _x to the exponent + _x to the exponent.
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We will see why this matters later on, especially in our next lesson, where we will talk about roots and zeroes of polynomials.
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But for now, it is really important to understand how we get somewhere from (x + 1)(x + 2) into x² + 3x + 2.
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This is probably going to be a bit of a review for most of you; but it is good to understand why this is happening,
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as opposed to just being able to do it mechanically.
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So, let's look at what is making it up.
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The thing this comes from is the distributive property, which says how multiplication interacts with parentheses.
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If something multiplies against parentheses, it distributes to every term that is separated by addition or separated by subtraction.
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For example, if we have a(b + c), then the a gets distributed onto the b, and the a gets distributed onto the c.
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So, we get ab + ac; that is how distribution works.
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How is that connecting to FOIL-ing things--how is it connected to different multiplicative factors for polynomials?
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Well, our distributive property is a(b + c) becomes ab + ac.
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From this property, we can use that on two different things in parentheses.
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We can distribute parentheses onto other parentheses; and the most basic form with two binomials,
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which is to say two things with two terms--we have the FOIL method.
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For example, if we have (a + b)(x + y), we can think of (a + b) as just being a block.
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So, like a is a block in our top example up here, we can think of (a + b) as being a block down here.
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(a + b) goes onto x, and (a + b) goes onto y; so we get (a + b) times x and (a + b) times y.
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Then, we turn right around, and we distribute in the other direction.
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We take x, and we distribute that onto the a and onto the b; and we take y, and we distribute that onto the a, and distribute that onto the b.
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And so, we get ax + bx, and then ay + by.
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Now, what does FOIL mean? FOIL is a mnemonic to help us remember the order of multiplication: Firsts, Outers, Inners, Lasts.
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Let's see how that comes to be; that end would be this way, where it is (a + b)(x + y).
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We would do the firsts; we would do a and x (those are the first things); so we would get ax.
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And then next, we would do the outers; a is on the outside, and y is on the outside (the outer part of our parentheses); we get ay + ay.
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b times x would be our inners, the things on the inner part of the parentheses...b times x.
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And then, b times y would be our lasts, because they are the last thing in each of our parentheses; plus b times y.
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And we see that these two things are exactly the same thing; it is just reordered.
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So, the distributive property and FOIL have the same thing going on here.
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It is just a way of being able to say, "How is this going to multiply? How is it going to distribute onto the other thing?"
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The way that we are making this FOIL method is two distributions, one after another.
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But when we are actually using the distributive property to multiply out polynomial factors,
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we probably want to think in terms of this first term, times the other terms inside,
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and then the second terms times the other terms inside, and then the third term, and so on, and so forth, and so on.
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This idea can expand into working on much longer parentheses than just two terms inside of it.
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So, instead of just using binomials, we could have something like (x² + 2x + 2)(3x² - x).
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So now, our first one has three terms, as opposed to just two.
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But the same method still works: we can have x² times 3x², and then x² times -x.
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Next, we will do 2x times 3x², and then 2x times -x.
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And then finally, we will do 2 times 3x², and 2 times -x; great.
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Each term in the parenthetical group multiplies all of the terms in the other parenthetical group.
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We have x² multiplying against 3x², and then multiplying against -x.
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So, each term in the parenthetical group--one of the things in our parentheses--multiplies all of the terms in the other parenthetical group.
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We start with factors, and we multiply them out; when we do that, it is called expanding.
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What we just saw here is called expanding.
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When expanding, we are normally expected to simplify.
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I didn't simplify this one, because we don't really want to get into having to do that right now.
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But we could simplify it pretty easily at this point.
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We would multiply things out; we would get x² times 3x² (becomes 3x⁴).
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And then, we would do that with all of the other ones, and eventually we could add like terms together.
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And we could simplify this into one of our general-form polynomials of _x to the exponent + _x to the exponent + _x to the exponent.
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We can get it back into that general form.
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Expanding is also sometimes called FOIL-ing; now, this is technically incorrect for larger factors,
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because remember: FOIL is based off of that mnemonic: Firsts, Outers, Inners, Lasts.
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So, that requires it to be 2 and 2 (two binomials put together).
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But when people say this, we still know what they mean; FOIL-ing just means...it is another way of saying "expanding."
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So, when somebody says "FOIL these polynomials" or "expand these polynomials," they are really getting across the same idea.
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Use the distributive property; simplify it.
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The reverse process, taking a polynomial and breaking it up into those multiplicative factors, is called factoring.
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So, when we have this large, general-form polynomial, and we break it into those pieces,
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like (x² + 2x + 2) and then (3x² - x), that is breaking it into the multiplicative factors; so we call it factoring.
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The long-term behavior of a polynomial is determined by the term that has the largest exponent.
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Other terms can have an effect; but their effect will become less and less noticeable as x approaches either positive or negative infinity.
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Basically, as x goes very far in either direction (either to the right or to the left),
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it is going to end up being the case that the polynomial will be controlled
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by whichever exponent is largest--the term that has the largest exponent.
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Why is this the case? Well, let's consider: if we have x, x², x³, x⁴, and x⁵,
00:18:17.900 --> 00:18:23.600
and we plug in different values for x, when we plug in 1, they end up pretty much all being the same.
00:18:23.600 --> 00:18:25.700
1, 1, 1, 1, 1...they are all exactly the same.
00:18:25.700 --> 00:18:28.000
We get nothing but the same thing out of each of them.
00:18:28.000 --> 00:18:34.600
But if we plug in something different, like 2, we start to see differences come up: 2, 4, 8, 16, 32.
00:18:34.600 --> 00:18:40.700
Of course, the differences aren't very large yet; but as the numbers get larger and larger that we are plugging in,
00:18:40.700 --> 00:18:52.800
5, 25, 125, 625, 3125...the difference between x² and x⁵ is now 3100.
00:18:52.800 --> 00:19:03.300
And if we just get up to x as 10 (plug in 10 for x), we get 10, 100, 1000, 10000, 100000...
00:19:03.300 --> 00:19:07.500
massive differences between x⁵ and x², or x⁵ and x.
00:19:07.500 --> 00:19:14.800
Even the difference between x⁴ and x⁵ is a difference of 90,000.
00:19:14.800 --> 00:19:22.000
And we are only at x = 10; clearly, x⁵...if we place all of these side-by-side...is going to be the massive winner.
00:19:22.000 --> 00:19:27.100
It is going to have huge amounts of control; it is going to contribute so much more to what the value will end up being
00:19:27.100 --> 00:19:30.400
than either x, x², x³, or x⁴.
00:19:30.400 --> 00:19:33.500
None of those are going to be nearly as important as x⁵.
00:19:33.500 --> 00:19:40.400
So, as x becomes very big (positive or negative), the polynomial will be controlled by whichever term has the largest exponent.
00:19:40.400 --> 00:19:46.700
The term that has the largest exponent--in this case, when we compared these 5, it would be x⁵.
00:19:46.700 --> 00:19:50.000
Whatever has the largest exponent is going to end up taking over.
00:19:50.000 --> 00:19:58.500
Even if it has a really, really tiny coefficient in front, like 0.0001 times x⁵, that will eventually get cracked.
00:19:58.500 --> 00:20:04.900
As x⁵ becomes larger and larger and larger, and we plug in fairly large x, like, say, 10000,
00:20:04.900 --> 00:20:10.300
it will be able to knock out that coefficient and still be more important than x⁴, x³, x², x.
00:20:10.300 --> 00:20:14.200
So, the only thing that really matters is which one has the largest exponent.
00:20:14.200 --> 00:20:22.300
Once you can figure out that, you know which one is going to be in control of the function at the extreme values of ±∞.
00:20:22.300 --> 00:20:24.400
One other thing can have an effect, though.
00:20:24.400 --> 00:20:29.500
The leading coefficient is very important, because it is going to be able to flip it.
00:20:29.500 --> 00:20:35.800
So, the largest exponent is the term that determines things; the term with the largest exponent determines what will happen.
00:20:35.800 --> 00:20:39.000
But the coefficient on that term will also matter.
00:20:39.000 --> 00:20:45.500
If the coefficient is positive, it behaves normally; but if the coefficient is negative, it is going to flip the term.
00:20:45.500 --> 00:20:48.400
What do I mean by that? Well, let's look at x².
00:20:48.400 --> 00:20:56.000
x² has a normal parabola arc like that; but if we have -x², it is going to flip it.
00:20:56.000 --> 00:20:59.100
So, with x², we end up going up on the left and up on the right.
00:20:59.100 --> 00:21:04.100
But with -x², we end up going down on the right and down on the left.
00:21:04.100 --> 00:21:07.700
So, it is going to be down on both sides, because the negative is flipping it.
00:21:07.700 --> 00:21:16.500
This leading term, whether it is a plus or a minus in front, is going to have control over what happens.
00:21:16.500 --> 00:21:21.400
Either we are doing things the normal way, or we are going to flip to the opposite of that.
00:21:21.400 --> 00:21:28.300
So, when a polynomial is in standard form (which is to say that the largest exponent is in the front), we call this the leading coefficient test.
00:21:28.300 --> 00:21:34.500
By knowing what the leading coefficient is and the degree of the polynomial, we will be able to know what the long-term behavior is.
00:21:34.500 --> 00:21:42.600
All you need to know to use the leading coefficient test is the degree of the polynomial and the sign of the leading coefficient,
00:21:42.600 --> 00:21:47.000
which is going to be either plus or minus (or negative, technically).
00:21:47.000 --> 00:21:50.600
We know what its long-term behavior will be like; we will see some pictures on the next one.
00:21:50.600 --> 00:21:55.100
Long-term behavior--what do we mean by that? That is what happens as x gets very big--
00:21:55.100 --> 00:21:59.900
as x goes out to plus or minus infinity, as it gets very, very far away.
00:21:59.900 --> 00:22:08.300
We haven't really determined what it means by very, very far away; but it is just eventually, in the long run, how things will behave.
00:22:08.300 --> 00:22:10.500
Let's look at some pictures to understand what this means.
00:22:10.500 --> 00:22:14.800
So, for the leading coefficient test, if we have an even degree (which is a polynomial
00:22:14.800 --> 00:22:25.300
where the leading exponent is going to be even, like x², x⁴, x⁶, x⁸, etc.),
00:22:25.300 --> 00:22:31.400
then if the coefficient is positive, on the right and on the left, we are going to be going up,
00:22:31.400 --> 00:22:37.100
because, when we plug in a very large positive number, it is going to still stay a very large positive number.
00:22:37.100 --> 00:22:44.100
If we plug in a very large negative number, then that even exponent will flip it to being positive; so we will still be going up.
00:22:44.100 --> 00:22:49.100
On the other hand, if we have a coefficient that is negative, then when we plug in a very large one,
00:22:49.100 --> 00:22:52.200
we will get a very large number out; but it will then get flipped to going negative.
00:22:52.200 --> 00:22:55.800
If we plug in a very large negative number, then it will get flipped to positive.
00:22:55.800 --> 00:22:58.800
But once again, the negative coefficient will hit it; and so it will go down.
00:22:58.800 --> 00:23:03.100
So, for an even degree with a positive coefficient, both the left and the right side go up.
00:23:03.100 --> 00:23:08.700
If we are an even degree with a negative coefficient, both the right and the left side go down.
00:23:08.700 --> 00:23:16.400
An odd one, though (that is to say, something like x¹, x³, x⁵, x⁷,
00:23:16.400 --> 00:23:22.400
and so on and so on)...if the coefficient is positive, then as we go very far to the right,
00:23:22.400 --> 00:23:27.500
we are going to go up; we plug in a very large number, and we will get a very large positive number out of it.
00:23:27.500 --> 00:23:35.100
But if we plug in a very large negative number, it has an odd exponent; so x³...
00:23:35.100 --> 00:23:42.600
-2 plugged into x³ is -2 times -2 times -2; three negative signs means we are left with a negative sign; so we would get -8.
00:23:42.600 --> 00:23:46.100
So, it starts to go down as it goes negative and negative.
00:23:46.100 --> 00:23:52.100
On the other hand, if we had a negative coefficient, then we would end up flipping that.
00:23:52.100 --> 00:23:57.400
As we plug in very large positive numbers, they will get flipped down to going in the negative way.
00:23:57.400 --> 00:24:00.000
And if we plug in a very large negative number, it will come out negative;
00:24:00.000 --> 00:24:04.700
but then it will get flipped by that coefficient, and it will go positive; it will go up; great.
00:24:04.700 --> 00:24:09.700
So, the leading coefficient test is: if we know it is an even and a positive, it is going to be up on both sides.
00:24:09.700 --> 00:24:13.500
If it is an even, and it is a negative in front, then it is going to be down on both sides.
00:24:13.500 --> 00:24:18.400
Odd and positive is going to be down on the left, up on the right.
00:24:18.400 --> 00:24:24.000
And odd with a negative is going to be down on the right, up on the left.
00:24:24.000 --> 00:24:27.200
So, just keep those pictures in mind, and think of flipping.
00:24:27.200 --> 00:24:32.400
Now, notice that in the middle, we have these dashed lines; and what those dashed lines say
00:24:32.400 --> 00:24:35.600
is that we don't have any idea what the middle part is going to look like.
00:24:35.600 --> 00:24:43.900
The leading coefficient test only tells us what happens on the extremes--on the far left and the far right.
00:24:43.900 --> 00:24:47.700
What is going to happen eventually, one day, in the long term?
00:24:47.700 --> 00:24:50.600
But what happens in the middle--that is going to depend on the specific thing.
00:24:50.600 --> 00:25:01.100
It could be very interesting; it could be not that interesting; we don't know what it is going to be until we get at specific function that we are looking at.
00:25:01.100 --> 00:25:02.900
Then, we can figure out what it is going to be exactly.
00:25:02.900 --> 00:25:13.000
The leading coefficient test just tells us what is going to happen in the long term, to the very far right and the very far left--those portions.
00:25:13.000 --> 00:25:14.300
All right, we are ready for some examples.
00:25:14.300 --> 00:25:22.100
What is n, the degree, for 2x⁴ - 8x³ + 2⁵x - 19?
00:25:22.100 --> 00:25:32.300
Remember, the degree is the largest exponent on a variable.
00:25:32.300 --> 00:25:38.800
We go through; we look at all of our variables; and we see that this is the largest exponent on any of our variables.
00:25:38.800 --> 00:25:44.600
We might notice this 2⁵; but then we remember that it has to be a variable.
00:25:44.600 --> 00:25:49.100
So, the 2⁵ doesn't get considered; and so, x⁴ is the case.
00:25:49.100 --> 00:25:55.300
n is just our degree for a polynomial; so we have n = 4; and what is a<font size="-6">n</font>?
00:25:55.300 --> 00:26:00.900
Remember, the first one was an here; and then a₃ goes with the x³.
00:26:00.900 --> 00:26:05.100
And then, a₂ would go with x²; but where is that?
00:26:05.100 --> 00:26:18.200
First, a<font size="-6">n</font> is 2, which is also the exact same thing as a₄, because we have n as 4, so a₄ = 2.
00:26:18.200 --> 00:26:25.000
What is a₃? Well, what is the coefficient for x³? That is 8.
00:26:25.000 --> 00:26:31.200
What is the coefficient for the x²? We look at this, and we realize that that didn't show up at all.
00:26:31.200 --> 00:26:37.700
But we could rewrite this as 2x⁴ - 8x³ + 0x²,
00:26:37.700 --> 00:26:43.300
because x² never showed up, so it must have been taken out by something; it has been taken out by this 0.
00:26:43.300 --> 00:26:54.800
Plus 2⁵x, minus 19...so if that is the case, then it must be that it is a₂ = 0.
00:26:54.800 --> 00:27:00.800
The plugging in for a₂ must be 0, because it has to be able to take out that x² term.
00:27:00.800 --> 00:27:05.300
Then, from there, we just continue: a₁ is equal to 2⁵;
00:27:05.300 --> 00:27:12.700
and finally, our last one is a₀ at the very end; a₀ equals -19.
00:27:12.700 --> 00:27:17.100
So now we see what all of the coefficients are; we know what the degree is; great.
00:27:17.100 --> 00:27:24.200
The second example: Expand and simplify this expression; we have (x - 2)²(x³ - x + 3).
00:27:24.200 --> 00:27:30.800
The first thing we have to do is realize that (x - 2)² is just the same thing as (x - 2)(x - 2).
00:27:30.800 --> 00:27:37.400
If I have smiley face squared, then that is the same thing as smiley face times smiley face.
00:27:37.400 --> 00:27:41.200
If I have (x - 2)², then that is just (x - 2)(x - 2).
00:27:41.200 --> 00:27:46.700
Then, x³ - x + 3: let's start on the left and work our way to the right.
00:27:46.700 --> 00:28:03.700
(x - 2)(x - 2); well, that will get us x² (x times x) - 2x - 2x -2(-2) (becomes + 4).
00:28:03.700 --> 00:28:08.300
And then, x³ - x + 3...I haven't really worked with that yet.
00:28:08.300 --> 00:28:17.700
Let's simplify the left side first: x - 2x - 2x + 4...sorry, not x times x; x times x becomes x²; sorry about that.
00:28:17.700 --> 00:28:27.200
We have x² - 2x - 2x + 4; x² - 2x - 2x becomes x² - 4x, as we combine like terms; + 4.
00:28:27.200 --> 00:28:32.300
Then, times the quantity x³ - x + 3.
00:28:32.300 --> 00:28:35.400
All right, let's use different colors for the various pieces we have here.
00:28:35.400 --> 00:28:44.100
x² times x³ becomes x⁵; x² times -x becomes -x³;
00:28:44.100 --> 00:28:48.300
x² times positive 3 becomes + 3x².
00:28:48.300 --> 00:28:53.300
The next color is for -4x; that was our x² portion.
00:28:53.300 --> 00:29:01.100
-4x we will do in blue; so -4x times x³ will become -4x⁴.
00:29:01.100 --> 00:29:11.100
-4x times -x becomes positive 4x²; and then, -4x times positive 3 becomes -12x.
00:29:11.100 --> 00:29:17.500
The final one we will do in green; 4 times x³ becomes + 4x³;
00:29:17.500 --> 00:29:23.100
4 times -x becomes -4x; 4 times 3 becomes + 12.
00:29:23.100 --> 00:29:24.900
Great; now we have to simplify this.
00:29:24.900 --> 00:29:29.400
Now, this isn't too difficult to simplify, but it is easy to get lost.
00:29:29.400 --> 00:29:36.900
Each of the steps that we are about to do is pretty easy; the hard part is making sure we don't accidentally have any tiny missteps as we work through this.
00:29:36.900 --> 00:29:40.200
So, I would recommend checking and doing them by exponent.
00:29:40.200 --> 00:29:42.900
The first thing we will do is look at all the x⁵'s.
00:29:42.900 --> 00:29:47.300
We see that there are no other x⁵'s, so we just bring it down; we have x⁵,
00:29:47.300 --> 00:29:52.100
and then we will cross this out, so that we don't accidentally see it again, and don't accidentally end up trying to use it again.
00:29:52.100 --> 00:29:56.000
Next, we have x⁴'s; where are our x⁴'s? We have -4x⁴.
00:29:56.000 --> 00:30:01.000
Do we have any other x⁴'s? We look through it; no, we don't have any other x⁴'s.
00:30:01.000 --> 00:30:07.100
So, we bring that down; -4x⁴; and then we cross it out, so we don't accidentally try to use it again.
00:30:07.100 --> 00:30:12.300
Next, let's look for our x cubeds; we have an x cubed right here--anywhere else?--yes, we do; we have another x cubed here.
00:30:12.300 --> 00:30:19.100
So, we bring those together: -x³ + 4x³ becomes + 3x³.
00:30:19.100 --> 00:30:25.600
-1 + 4...we get + 3x³; and then we cross those out.
00:30:25.600 --> 00:30:33.500
Next are 3x² and 4x²; there are no other x squareds; 3x² + 4x² becomes 7x².
00:30:33.500 --> 00:30:41.500
We cross those out; next are x's; -12x - 4x; combine those together, and we get -16x.
00:30:41.500 --> 00:30:46.300
Take those out; and + 12; there we are.
00:30:46.300 --> 00:30:52.000
Now, you don't have to do this method of saying, "Here are my x⁵'s; here are my x⁴'s" and so on,
00:30:52.000 --> 00:30:53.900
and so on, and then crossing them out as you go.
00:30:53.900 --> 00:30:56.500
But this is a great way to make sure you don't accidentally make a mistake.
00:30:56.500 --> 00:31:00.800
It is easy, when you are working with this many terms and trying to put them together and simplify,
00:31:00.800 --> 00:31:03.700
to make one tiny mistake and lost the entire problem because of it.
00:31:03.700 --> 00:31:06.900
So, it is a good idea to have some method of being able to follow your work and make sure
00:31:06.900 --> 00:31:11.100
you don't accidentally try to do the same thing twice, or completely miss a term.
00:31:11.100 --> 00:31:18.900
All right, the next one: Give an example of a quadratic trinomial, a cubic monomial, and a linear binomial.
00:31:18.900 --> 00:31:30.100
Quadratic trinomial: remember, quadratic meant degree 2; and then, trinomial meant three terms.
00:31:30.100 --> 00:31:39.700
A cubic monomial is a degree 3 (cubic means degree 3); monomial...mono- means single, like monorail,
00:31:39.700 --> 00:31:46.900
a train track with one rail (not really a train anymore); a monomial is one term.
00:31:46.900 --> 00:31:58.700
And then finally, linear is degree one; and binomial is two terms (bi- like bicycle); great.
00:31:58.700 --> 00:32:07.200
So, if we want to give an example of this, we just need something that is degree 2 and 3 terms.
00:32:07.200 --> 00:32:12.400
If it is degree 2 and it has 3 terms, then we are going to have something that has x² at the front;
00:32:12.400 --> 00:32:18.100
and it has to have blank spots for a total of three things.
00:32:18.100 --> 00:32:22.300
Now, we can't have zeroes show up in these, because then it would disappear and we wouldn't have a term there.
00:32:22.300 --> 00:32:29.500
We will have to put in something; so let's call it 5x² + 3x, and we will make it -17.
00:32:29.500 --> 00:32:32.600
You could plug anything into these blanks, and the answer would still be correct.
00:32:32.600 --> 00:32:38.600
5x² + 3x - 17; there is our quadratic trinomial.
00:32:38.600 --> 00:32:43.000
Next, we do a cubic monomial; we know it has to be degree 3.
00:32:43.000 --> 00:32:48.900
Degree 3 means it has to be x³; and it is only one term, so there is going to be a blank in front of the x³.
00:32:48.900 --> 00:32:57.100
But we are not allowed to have any other blank things, because if we did, then we would have more than one term.
00:32:57.100 --> 00:33:02.000
We are only allowed to have one term; so all of that gets taken right out--it disappears.
00:33:02.000 --> 00:33:10.600
We have just _x³; we plug whatever we feel like in...I feel like -47, so we get -47x³.
00:33:10.600 --> 00:33:17.800
Great; the final one--we have a linear binomial: a binomial has to have two terms, and linear is degree 1.
00:33:17.800 --> 00:33:23.700
So, we have x¹, with some blank in front of it, plus blank, _x + _.
00:33:23.700 --> 00:33:25.700
What goes in those blanks? Whatever we feel like.
00:33:25.700 --> 00:33:29.000
We are not allowed to have any other blanks, though, because then we would have more than two terms.
00:33:29.000 --> 00:33:32.800
Also, we can't have any more blanks, because we are linear, and that is the most that we have there.
00:33:32.800 --> 00:33:42.100
So, _x + _...let's put in 1 for the x and -7 for the constant; so we have x - 7.
00:33:42.100 --> 00:33:47.500
Great; the last thing--explain why it is impossible to have a linear trinomial.
00:33:47.500 --> 00:33:51.800
So, if you are going to have a linear trinomial, let's see what that structure has to be.
00:33:51.800 --> 00:33:55.900
Well, if we are linear, we know that x is going to be at the front.
00:33:55.900 --> 00:34:01.300
And so, if we do the normal structure that we have for polynomials, it will be _x + _.
00:34:01.300 --> 00:34:05.800
But if it is a trinomial, "trinomial" means we have to have three terms.
00:34:05.800 --> 00:34:10.800
So, if we try to force on a third term, we would have to have _x².
00:34:10.800 --> 00:34:17.800
We already have _x + _, so the only way to go is to go to the left; we have to have higher and higher exponents.
00:34:17.800 --> 00:34:25.100
So, _x²...all of a sudden, now we are a trinomial, but we are not linear anymore.
00:34:25.100 --> 00:34:28.300
So, it means that we can't have both of these things at the same time.
00:34:28.300 --> 00:34:33.300
We can't both be linear and have a third term; otherwise we would have to have x²,
00:34:33.300 --> 00:34:36.100
at which point we wouldn't be linear anymore; we would be quadratic.
00:34:36.100 --> 00:34:41.900
So, it is going to be one or the other; you can't be both a trinomial and a linear function.
00:34:41.900 --> 00:34:48.200
The final example: What is the degree of y = (-2x² + 4)^407?
00:34:48.200 --> 00:34:53.900
Now, you see this at first, and you might get scared, because you think, "I can't possibly expand 407 times--I can't do that!"
00:34:53.900 --> 00:34:57.200
But don't worry; all they asked for was the degree.
00:34:57.200 --> 00:35:06.800
So notice: if I have (x² + 3)(x⁵ + 48), do I have to look at anything else
00:35:06.800 --> 00:35:09.800
to figure out what the degree is going to be, other than the front parts?
00:35:09.800 --> 00:35:14.600
No, because I know only the x² and the x⁵ are going to come together to make x⁷.
00:35:14.600 --> 00:35:20.100
And there is going to be other stuff; but I know I can't get any higher exponents out of this than the x⁷.
00:35:20.100 --> 00:35:23.200
It is going to be the leading term that will have the highest exponent.
00:35:23.200 --> 00:35:26.000
It is going to be the exact same thing on this one.
00:35:26.000 --> 00:35:32.500
It is going to be that -2x²; it is a question of how many times -2x² hits -2x².
00:35:32.500 --> 00:35:35.800
That is the only thing that is going to be able to really bring increases of the degree.
00:35:35.800 --> 00:35:41.200
There is going to be a whole bunch of other stuff; but we are not concerned with it, because all that they asked for was the degree.
00:35:41.200 --> 00:35:47.800
It is going to be -2x² raised to the 407, plus other stuff.
00:35:47.800 --> 00:35:59.900
But we don't care about the other stuff: -2x² to the 407...we distribute that...-2^407(x²)^407.
00:35:59.900 --> 00:36:05.300
So, if we have 407 x², then it is x² times x² times x² times x²...
00:36:05.300 --> 00:36:14.300
So, it is going to be the same thing as x^2(407), because they are going to iterate that many times; it is going to hit that many times.
00:36:14.300 --> 00:36:30.600
So, we have (-2)^407, times x^2(407); (-2)^407x^814.
00:36:30.600 --> 00:36:37.800
So, our degree is n = 814; that is our degree for this polynomial.
00:36:37.800 --> 00:36:45.900
Now, as x goes very far to the left (x goes to -∞), will y go up or down (y approaches +∞ or y approaches -∞)?
00:36:45.900 --> 00:36:49.600
And then, what about as x goes very far to the right--as x goes to positive infinity?
00:36:49.600 --> 00:36:54.000
So, to do that, we need the leading coefficient test.
00:36:54.000 --> 00:36:59.300
At this point, we already know what the degree of this polynomial is.
00:36:59.300 --> 00:37:06.400
This polynomial is n = 814; so it is an even-degree polynomial.
00:37:06.400 --> 00:37:13.200
Now, we want to figure out what our leading coefficient is; is it positive or negative--plus or minus?
00:37:13.200 --> 00:37:23.600
We do that: -2 to the 407, times x to the 814...well, if it is a negative raised to an even number, they will all get canceled out.
00:37:23.600 --> 00:37:30.200
If it is a negative raised to an odd number, one of them remains, because it will end up getting to stay around.
00:37:30.200 --> 00:37:34.800
All of the even part will get canceled out, but that odd is an extra +1, so it stays around.
00:37:34.800 --> 00:37:45.600
So, we will get -2^407x^814; that means we have a negative sign right here.
00:37:45.600 --> 00:37:56.600
So, by the leading coefficient test, we have negative and even; negative and even means an even one.
00:37:56.600 --> 00:38:03.000
Even normally goes in the same way that a parabola goes; it cups up, normally (even at positive).
00:38:03.000 --> 00:38:07.200
But even at negative will flip that cupping shape, and we will get that.
00:38:07.200 --> 00:38:12.200
Now of course, we don't actually know what is in the middle; all we know is the extremes,
00:38:12.200 --> 00:38:15.500
because that is all we were guaranteed from the leading coefficient test.
00:38:15.500 --> 00:38:18.600
But that is all we have to figure out, because it is as x approaches negative infinity.
00:38:18.600 --> 00:38:24.000
So, from this, we see, even as it goes negative, that we go down on the left and down on the right.
00:38:24.000 --> 00:38:32.000
So, as x approaches negative infinity--as x goes very far to the left--we are going to approach y going to negative infinity.
00:38:32.000 --> 00:38:41.400
As x goes very far to the right (x goes to infinity), we are going to get y going to negative infinity, once again.
00:38:41.400 --> 00:38:44.100
All right, great--the leading coefficient test should be able to figure that out.
00:38:44.100 --> 00:38:46.200
All right, we will see you at Educator.com later.
00:38:46.200 --> 00:38:50.700
Next time, we will look at roots and zeroes of polynomials and get a really good understanding of how these things are working.
00:38:50.700 --> 00:38:41.000
All right--goodbye!