All right so hi everyone, let's start talking about polynomials and let's start with a basic definition. We've seen this before but so hopefully this is a little bit of review but we talk about n being a non negative integer. So we wanted to be non negative, that means it can be 0 and it can be positive, it just can't be negative. So we want this to be an integer, these are numbers like 1,2,3,4, 5. And we also want to pick a bunch of numbers a1, a2 dot dot dot a n, pick any of your n favorite real numbers. And then we're going to put these all together to create what's called a polynomial. And in its most general form we have a to the n, x to the n plus a to the n minus 1, x to the n minus 1 plus dot dot dot a1x plus a0. This is a big fancy complicated way to say a polynomial but of course, you know it when you see it. The ones that you normally see are the constant polynomials. So for example, just when I have the a0 term, this is like, I don't know how about y equal 7? So all the original coefficients in front of 0 and the last number is constant. We can call it y equals 7 you can also write of course is f of x equals 7. You've also seen linear equations, this is just when I have the first and second terms here from the right, so how about 3x plus 4. These are your lines, we have seen when I go one more out you can have a quadratic, so how about 4x squared plus 7x plus 2. Is a quadratic polynomial. So polynomial just generalizes these terms, these linear terms, these quadratic expressions that we've seen before to allow for anything. And now we're just going to allow for like y equals x to the 10 plus 7x to the 9 plus. You get the idea how about 3x plus 2 and you simplify it up, but you're allowed to have as high a degree. We call the degree the exponent of the leading term here, so remember when you have only one term, you're call it a constant. This is the constant term, when you have one expression, one variable exponent. The degree is 1 your linear, if your highest degree when your highest exponent is 2. This is a quadratic and polynomial is with one, two or three terms, they're called monomial, binomial or trinomial respectively. So this one has three, so it's a trinomial, we usually write the turns of polynomial in a single variable so that the exponents are in descending order. Notice I've written each one of these in descending order from left to right, you can certainly move them around. But this is the traditional way to write it and one of polynomial is written in this manner. The coefficient of the first term is called the leading coefficient. So in this example here leading coefficient is 3, leading coefficient is 4 and leading coefficient here is 1. The degree of the polynomial, this is the highest exponent that appears so degree 10, degree 2, degree 1. Degree 1 are called linear, degree 2 are called quadratic, some of them have named some of them don't if your degree 3. So if I wrote something like with the 3, that would be called a cubic and perhaps equally is important to know what is a polynomial. It's equivalent to know what is not a polynomial, so how do I break a polynomial? Well, how about y equals 1 over x, this is x to the minus 1 and remember we said the integer has to be non negative. So that's not allowed, other things that are not allowed, how about the square root of x? Which is like x to the one half again that's not an integer although it is not negative. And of course all your other functions like sine, cosine, elog, exponential those are all not as well. So polynomial is a very specific type of function, you can think of polynomial is is sort of generalizing a number. And so of course you can add and subtract them, we're just going to do a couple of these by examples. But the general idea is to find and combine like terms, so if I had the quadratic polynomial of negative x squared plus 3x. I put the parenthesis and I subtracted from it, x squared minus 5x plus 1. So I have two quadratic that subtract, let's clean this up a little bit. So as usual bring the minus sign into all the pieces we distribute, that's called distribution. So we have minus x squared plus 3x, here we go ready minus x squared plus 5x minus 1. And now we combine like terms, so I have a degree two term here and a degree two term there. So I get minus 2x squared, I have a linear term degree 1, 3x and 5x put them together of course you get plus 8x. And then the only constant I see is a minus 1, so my final answer is negative 2x squared plus 8 x minus 1. Challenge with polynomial of course, is that you can have many many terms, you just have to be careful. Let's do a higher degree term, let's do 3x cubed minus x plus 5 and let's add that to negative 8x cubed plus 3x minus 9. Again the key here is to combine and find like terms, you sort of look for all the similar terms. I see a 3x cubed and a minus 8x cubed, so it's like I combine the coefficients 3 minus 8. That's negative 5x cubed, negative x plus 3, so that's plus 2x and then 5 minus 9 my constant, well that's just good old minus 4. Let's do another example, what if I had minus 3x squared minus x plus 1 minus parentheses x squared minus 9. These aren't incredibly difficult just to be careful of these minus signs, always use parentheses just to be safe. So we bring in the minus sign, this becomes minus 3x squared minus x plus 1 minus x squared plus 9. Oftentimes I see people do these in their head, they just make silly mistakes. I make silly mistakes sometimes, so check my work always look for combining like terms. I see a degree too, so minus 3 minus 1 is minus 4x squared and then I see only x here. There's nothing to combine it with, that's okay you're not always going to have things that have pairs or other parts. So we just bring it along for the ride and then 1 and 9 of course that's 10 and you get back another polynomial. So there's two skills here going on, one is to combine using arithmetic, you can add subtract or do multiplication a second. But the other bigger thing to kind of keep in mind is that whenever you add or subtract two polynomial, you get back another polynomial. That's true for addition, is true for subtraction let's see multiplication, it's going to turn out to be true as well. So let's look at minus 3 x times 2x minus 3, so as always we distribute, bring it in. Just watch your minus signs, minus 3 times 2 deal with the constants first this minus 6 and then x times x. This is good old x squared add the exponents as you go and then bring the minus 3. And again two negatives make a positive and we get positive 9x. We're going to do one more, what if I had x squared minus 2x squared minus 3. Now I'm going to do this a long way and I'll show you a shorter way to do in a second. Technically what I'm going to do is I'm going to bring both these in and distribute. So that said I have x squared minus 2 times x squared, right I multiply the whole first expression by the first term and then I have minus 3. Maybe I'll bring that up front times x squared minus 2. Like this is okay if you know the shortcut just hold on a second, we'll do that next. So let's bring the x squared in, when I have x squared times x squared, I have good old x the fourth then minus 2x squared. And then if I bring the negative 3 in and you can bring it and left or right, doesn't matter. Multiplication is communicative minus 3x squared and then plus 6. Watch that plus 2 negatives make a positive and there's a little bit to combine here in the middle. So this is x to the fourth minus 5 x squared plus 6, this is our most simplified form after combining all like terms. But I want to show you that the idea of distribution holds even if you have more complicated polynomials. When you have the product of two binomials, that's just the polynomial with two expressions, so these are binomials. You might know this, perhaps you've heard this expression before, but you can also do this as a foil and we're going to look at that in the next slide. So let's do this one again, but we'll do FOIL and we can compare our answers. So FOIL stands for first outside inside a last, so let's Look at this example one more time. It only works if you have binomial, so it's a little bit of a shortcut in a very special case. However, it comes up so much that it's worth knowing, so FOIL says take the first ones, so this is your first F and start putting things together X squared times X squared is actually the force outside says let's go outside there's your o. So it's X squared, times -3. Watch the negative, you get -3 X squared. The first outside here comes your inside. I'm going to run our room here but inside -2 x squared. And then of course last -2 times -3 +6. It's just a shortcut so you don't have to distribute. And again it only works when you have two things times two things. So it stands for first outside Inside last it just tells you the order in which you can multiply after you do your first outside inside last of course you still need to simplify and you have to combine like terms that gives us -5 X squared +6. And you can compare this is a little shorter, right? It's two lines compared to the other one where it was the longer. But you get the same answer. So the foil method is quick. It's good. It's a good memory aid. It helps you things out. But you can you can certainly foil. Let's just do one more. How about we do (a+b) quantity squared? Okay. So I remember here when you have something like this, we do not bring the two and it is a common rookie mistake to try to bring the two in over edition. So I'm going to put this in red here. This is not a squared plus B squared. No, don't do that. I call it the high school dream here. Like you wish that were true. But that is absolutely not true. But I see a lot of people make that mistake. Just remember this is a+b times a+b. You are allowed to bring exponents in over multiplication. That's okay. You can certainly do that multiplication division, but you cannot bring it in over addition or subtraction. So do not bring in the exponents say, isn't it just a sport? No you have to foil, you have to foil this out. So be careful here. When you do this, first is a squared. Outside is a b. Inside is b a. But I'll just write as ab. So it looks similar and then last is b squared. So first outside inside last combine the two terms in the middle and you get a squared plus two a b plus b squared. This is our final answer notice if you had just mistakenly brought in the square, you would have lost that middle term and of course been just wrong. So just watch out for that. So this a squared plus b quantity squared, It's a squared plus two a b minus b squared. Be careful, be careful, be careful. Oftentimes you might need to rationalize the denominator. This means is a fancy way to say, get rid of square roots inside the denominator. It's a little bit of an algebraic trick. And there are times when you want to do this and there are times when you don't, but sometimes you get a fraction like this where there's a square root downstairs. Now remember this is an irrational number and there might be a time later on in your math career when you don't want numbers downstairs as a number itself, this is perfectly fine. You can certainly plug this into any calculator or not. But if you know a little bit about computers, rational numbers, a little messy. They get to floating point arithmetic. It's they're not the nicest things to work with all the time. So often we're going to rationalize the denominator. That's a fancy way to say multiply by one but not just anyone. We're going to do 2 plus route 2 over 2 plus root to appreciate this for a minute I am multiplying by a fraction with the same numerator and the same denominator. This fraction is a big fancy way to write the number one. So really I have not changed my number. Alright if I take this expression with it without a new fancy one, I'm going to get the same decimal representation back. But why would I do this this again? You don't see the benefits this into a little bit later. But this is a skill you want to sort of take away here, Multiplying fraction by fraction. Friendly reminder. How do you want to play fractions? You go across the top and you go across the bottom. So this top fraction becomes two times two plus route to over 2 -12 times two plus route to notice. By the way that I change the sign. When you rationalize a denominator, you do want to change the sign. So because I started with a negative, I'm going to put a positive on my product on the thing, I multiply by. If they had given me a positive, I would have switched it and made it a minus. The reason for that is because you want the difference on the denominator when you work this out, keep cleaning this up. So let's bring in the two to both pieces and you get two times two, which of course is four plus two route to So that's my numerator and now downstairs, here it comes. I have two binomial is multiplied together so of course, what am I going to do? I'm going to foil. Here we go. So let's do it carefully. First two times two got that. That's for outside becomes two times route to Not much I can do with that. I just put them together to root to and then inside becomes negative. Two times route to. That's nice. That's why we switch signs because what's going to happen is these two terms canceled and then last but not least is the outside negative times positive is negative and then root two times root two is two actually I'm going to bring that to back out for a second. You'll see why in one second. so I'm going to leave. It is Two times two plus or to use parentheses and my denominator is just 4 -2. That's two. So you can see why I brought the two back out because it will cancel. So my final cleaned up fraction at the end of the day is a much simpler expression two. Plus or two. And if you want you can check, go through both these numbers in the calculator. Use parentheses If you plug it in the first expression and you'll get the same exact decimal expression. They're exactly the same. But one just a little more cleaned up and this is the idea of rationalizing the denominator. Sometimes it just gives you nicer numbers to work with polynomial like numbers which you can divide. I can certainly take two divided by three. You start to get into remainders and it gets a little messy division of polynomial in general is a little messy and we're going to save the sort of messy expressions for another day. But just so you have the vocabulary that's associated with division of polynomials. We say that if the dividend P(x). And the divisor D(x) are polynomials. So any polynomial as you want such that D(x) is non zero Is usually can't divide by zero. Then there exist unique polynomials Q(x) called the quotient and the remainder R(x) such that P(x)equals Q(x) times D(x) plus our r(x) where R(x) is zero or the degree of R (x) is less than the degree of D(x). This is a fancy way to say you can divide polynomial. It gets a little messy. If you want to look up expressions, if you want to go on your own, look up more polynomial, look up synthetic division or look up polynomial division. The cases that I want you to know for sort of our purposes are the relatively simple ones. Where if I did something like 8x to the fifth divided by 2x squared. I want you to be comfortable with this, I want you to realize this is a text of the 5th divided by two x squared is a mono meal divided by a mono meal. And then your algebra rules come into play, divide the numbers First, the constants, the coefficients eight divided by two is of course four. And then friendly reminder when you have X in the base and you're dividing, you subtract exponents, this is X to the five minus two which is X to the third. So final answer is four X cubed. We can do another one. What if I had X Cubed -8. So more Complicated Here Divided by X -2. What I'm looking for here is I don't want to go through the whole division algorithm. I want to look to factor this expression, I want to split this up somehow. Some way realise that too, I can write x cubed -8 as A factor of X -2, x squared plus two X plus four. Now that's not obvious. This is called the difference of Cubes eight of course being two cubed. But you can check that that's true. And so when you can factor, when you can break off a piece, that's kind of nice things cancel and you're left with X squared plus two X plus four. So some of these are going to be a little easier than others and always look for something to factor in something to cancel as you go through and practice your examples. Just to give you one more. How about x squared- 3x + 2 divided by x- 1? So I'm looking at this and saying, this is not monomials I have some expression here, can I factor, can I reverse foil my polynomial? How do I write this? So we have x -1. How about x + 3? Check that, foil that back. You do get back the original expression and now things cancel. And of course this is just good old, x + 3. Let's do an example now where we solve a polynomial equation. Let's try to find 6 x to the 5 + 24 x cubed = 0. Use algebraic skills for polynomial to see how to solve this. The first thing we look at, let's say, do these expressions have anything in common? Do they have any common terms? This is a degree 5 equation, also known as a quintic. And you could look at this and say, wait a minute. Well, 6 and 24, I can factor out 6 from both of the coefficients. And there's 5 xs here that are multiplied together and 3 xs is here. So I can certainly factor out 6 x cubed. When you factor that out, you're left with x squared + 4 is 0. When you have two terms together that multiply to give you 0, one of them has to be 0. So that's going to tell you that 6 x to the cubed equals 0, or x squared + 4 is 0. Now, 6 x cubed is 0, you can divide both sides by 6. You'll get x cubed is 0. What's the only number cubed that gives you 0? Well, that of course gives you x = 0. On the other side over here, we sort of have a choice to make here. We can say that there's no real solutions. If you want to get fancy, you can certainly introduce complex numbers, because you're going to get x squared is -4. There's no real number squared that gives you -4. If you know a little bit about complex numbers, again, you don't have to be too fancy for us in this case. You're going to get, take a square root of both sides, you get plus or minus 2i. I'll leave that as sort of good to know, but you don't actually have to know it for the stuff purposes of this course. But complex numbers are out there. And well, I guess you shouldn't be scared when they do pop up. Now let's look at the function x cubed- x squared- 9x + 9. And let's set it equal to 0 and see if we can solve for x. To solve this polynomial equation. One thing you might look at is in all four terms. They don't really have anything in common. There's only xs in the first three, there's 9 in the second two. That leads us to another sort of approach to solving these things. Instead of treating all four pieces as one, let's look at just the first two together and perhaps the second two together. And say, well they have things in common. So let's try to play around and factor out things of just the first two. When you look at this, this is called factor by grouping. You're almost like drawing the parentheses around each of the pieces. Here I got to be careful of that negative sign. But in the first two I can certainly factor out an x squared. Because both terms have an x squared. I'm left with x -1. In the second part, I can factor out of -9. I can factor out -9. And when I do that, I get x -1. Stare at that for a second. Just convince yourself with that minus sign, that if I bring the -9 back in, I do in fact add -9x + 9. And now all of a sudden you'd say, wait a minute. Now they do actually have something in common. There's an x- 1 that's common to both terms. So I can factor out the x- 1 and I'm left with x squared- 9. So this factor by grouping technique is really nice, especially if you have polynomial with many terms. x -1 is already simplified. And x squared- 9, well, that's (x -3) ( x + 3). And that's going to give me all of it equal to 0. So you set each piece equal to 0. You get x- 1 = 0, x- 3 =0, x + 3 = 0, right? If it's going to be 0, one of these things has to be 0. So the roots of this equation become, the solutions to this equation are 1, 3, and then -3. So we have three solutions to this equation. And that's okay, you can certainly have more than one solution. All right, so practice solving your polynomial equations. Look for these factoring techniques. Look for ways to rewrite or change the expression to a more simpler form to work with. Rationalize your denominators as needed. Have some fun doing these. Again, these are really good to really solidify your algebraic skills. Working with higher degree polynomial. Great job on this video. And we'll see you next time.
