And polynomial functions I would be looking at functions of X that represent polynomials of varying degrees, including cubics and quadratics and Quartics. We will look at some of the properties of these curves and then we will go on to look at how to deduce the function of the curve if given the roots. So first of all, what is a polynomial with a polynomial of degree add is a function of the form F of X. Equals a an X to the power N plus AN minus One X to the power and minus one plus, and it keeps going all the way until we get to. A2 X squared plus a 1X plus a zero and here the AA represents real numbers and often called the coefficients. Now, this may seem complicated at first sight, but it's not and hopefully a few examples should convince you of this. So, for example, let's suppose we had F of X equals 4X cubed, minus three X squared +2. Now this is a function and a polynomial of degree three. Since the highest power of X is 3 and is often called a cubic. Let's suppose we hard F of X equals X to the Power 7 - 4 X to the power 5 + 1. This is a polynomial of degree Seven. Since the highest power of X is A7, an important thing to notice here is that you don't need every single power of X all the way up to 7. The important thing is that the highest power of X is 7, and that's why it's a polynomial degree 7. And the final example F of X equals. Four X squared minus two X minus 4. So polynomial of degree 2 because the highest power of X is a 2. Less often called a quadratic. Now it's important when we're thinking about polynomials that we only have positive powers of X, and we don't have any other kind of functions. For example, square roots or division by X. So for example, if we had. F of X equals 4X cubed. Plus the square root of X minus one. This is not a polynomial because we have the square root of X here and we need all the powers of X to be positive integers to have function to be a polynomial. Second example. If we had F of X equals 5X to the Power 4 - 2 X squared plus 3 divided by X. Once again this is not a polynomial and this is because, as I said before, we need all the powers of X should be positive integers and so this 3 divided by X does not fit in with that. So this is not a polynomial. OK, we've already met some basic polynomials, and you'll recognize these. For example, F of X equals 2 is as a constant function, and this is actually a type of polynomial, and likewise F of X equals 2X plus one, which is a linear function is also a type of polynomial, and we could sketch those. So the F of X. And X and we know that F of X equals 2 is a horizontal line which passes through two on the F of X axis and F of X equals 2X plus one. Is straight line with gradient two which passes through one. On the F of X axis, stuff of X equals 2 and this is F of X. Equals 2X plus one. Important things remember is that all constant functions are horizontal straight lines and all linear functions are straight lines which are not horizontal. So let's have a look now at some quadratic functions. If we had F of X equals X squared, we know this is a quadratic function because it's a polynomial of degree two and we can sketch. This is a very familiar curve. If you have F of X on the vertical axis. An X on our horizontal axis. We know that the graph F of X equals X squared looks something like this. This is F of X equals X squared. So what happens as we vary the coefficient of X squared? That's the number that multiplies by X squared, so we had, for example, F of X equals 2 X squared. How would that affect our graph? What actually happens is you can see that all of our X squared values are the F of X values have now been multiplied by two and hence stretched in the F of X direction by two. So you can see we actually get. Occurs that looks something like this. And this is F of X equals 2 X squared. And likewise, we could do that for any coefficients. So if we had F of X equals 5 X squared. It would be stretched five times from the value it was when it was X squared, and it would look something like this. So this would be F of X equals 5 X squared. So this is all for positive coefficients of X squared. But what would happen if the coefficients were negative? Well, let's have a look. It starts off with F of X equals minus X squared. Now, compared with F of X equals X squared would taken all of our positive values and we've multiplied them all by negative one. So this actually results in a reflection in the X axis because every single value is now, which was positive has become negative. So F of F of X equals minus X squared will look something like this. So this is F of X equals minus X squared. Likewise, F of X equals minus two X squared will be a reflection in the X axis of F of X equals 2 X squared, and so we get something which looks like this. F of X equals 2 X squared 3 - 2 X squared and Leslie are going to look at F of X equals minus five X squared and you may well have guessed by now. That's a reflection of F of X equals 5 X squared in the X axis, which gives us a graph which would look something like. This. We F of X equals minus five X squared. And in fact, it's true to say that for any polynomial, if you multiply the function by minus one, you will always get the reflection in the X axis. I will have a look at some other quadratics and see what happens if we vary the coefficients of X as opposed to the coefficient of X squared. And to do this I'm going to construct the table. So we'll have RX value at the top. And the three functions I'm going to look at our X squared plus X. X squared plus 4X. And X squared plus 6X. And for my X file use, I'm going to choose values of minus 5 - 4 - 3 - 2 - 1 zero. I'm going to go all the way up to a value of two. Now for each of these functions, I'm not necessarily going to workout the F of X value for every single value of X. I'm just going to focus around the turning point of the quadratic, which is where the quadratic dips. So just draw some lines in on my table. On some horizontal lines as well. And now we can fill this in. So for RF of X equals X squared plus X, I'm just going to focus on minus 2 + 2. So when I put X equals minus two in here we get minus 2 squared, which is 4 takeaway two which gives Me 2. When we put X is minus one in we get minus one squared, which is one. Take away one which gives us 0. Report this X is zero in here we just got 0 + 0 which is 0. I put X is one in here. We've got one plus one which gives Me 2. And finally, when I pay taxes 2 in here, the two squared, which is 4. +2, which gives me 6 and in fact just for symmetry we could put X is minus three in here as well, which gives us minus 3 squared 9. Plus minus three. Sorry, which gives us 6. OK, for my next function, X squared plus 4X. I'm going to look at values from minus four up to 0. So if I put minus four in here, we get 16 takeaway 16 which gives me 0. Put minus three in here. We get 9 takeaway 12 which gives us minus three. For put minus two in here, we get four takeaway 8 which is minus 4. If I put minus one in here, we get one takeaway 4 which is minus three, and if I put zero in here 0 + 0 just gives me 0. And Lastly, I'm going to look at F of X equals X squared plus 6X. And I'm going to take this from minus five all the way up to minus one. So we put minus five in first of all minus 5 squared is 25. Take away 30 gives us minus 5. Minus four in here 16 takeaway 24 which gives us minus 8. For X is minus three in. Here we get 9 takeaway 18, which is minus 9. For X is minus two in. We get minus two square root of 4 takeaway 12, which is minus 8. And we put minus one in. We get one takeaway 6 which is minus 5. And you can actually see the symmetry in each row here, to show that we've actually focused around that turning point we talked about. So let's draw the graph of these functions. So in a vertical scale will need. F of X and that's going to take us from all those values minus nine up 2 - 6. So. If we do minus 9. Minus 8 minus Seven 6 - 5. For most 3 - 2 - 1 zero. 12345 and we can just about squeeze 6 in at the top. On a longer horizontal axis. We've gone from minus five to two. So we just use one 2. Minus 1 - 2 - 3 - 4. And minus 5. So first of all, let's look at our function F of X equals X squared plus X. We've got minus three and six. Which is appear. We've got minus two and two. Which is here. Minus one and 0. We've got zero and zero. One and two. And two 16. And you can see clearly that this is a parabola which, as we expected, we can draw a smooth curve. Through those points. So this is F of X equals X squared. Plus X. Next, we're going to look at the function F of X equals X squared plus 4X. And the points we had with minus four and zero. Minus three and minus three. Minus two and minus 4. Minus one and minus three. And zero and zero which is already drawn. And once again you can see this is a smooth curve is a parabola as we expected. And this is F of X equals X squared plus 4X. And the final function we're going to look at F of X equals X squared plus 6X. And so we've got minus 5 - 5. So over here minus 4 - 8. Which is down here. Minus three and minus 9. Minus 2 - 8. And minus one and minus five and once again you can see smooth curve in the parabola shape. And this is F of X equals X squared plus 6X. So we can see that as the coefficient of X increases from here to one to four to six, the curve the parabola is actually moving down and to the left. But what would happen if the coefficients of X was negative? Well, let's have a look and will do this in the same way as we've just on the previous examples. And we look at a table, and this time we'll look at how negative values of our coefficients of X affect the graph. So we look at X squared minus XX squared minus 4X. And the graph of F of X equals X squared minus 6X. Now as before, will. Put a number of values in for X, but we won't use. All of them, will only use the ones which are near to the turning point of the quadratic, but the values will put in the range will be from minus two all the way up to five. Four and five. I just put in the lines for our table. OK, so for the first one, F of X equals X squared minus X. We look at what happens when we substitute in X is minus 2. So minus 2 squared is 4 takeaway, minus two is the same as a plus two, which gives us six when X is minus 1 - 1 squared, is one takeaway minus one is the same as a plus one which just gives us 2. Zero is just zero takeaway 00. When we put one in, we've got 1. Take away, one which is 0. And two gives us 2 squared four takeaway two which gives us 2. And again, for symmetry will do access 3. So 3 squared is 9 takeaway, three is 6. So our second function F of X equals X squared minus 4X. We're going to look at going from zero up to five. And we put zero and we just get zero takeaway 0 which is 0. When we put X is one. We get one takeaway 4 which is minus 3. When we put X is 2, two squared is 4 takeaway eight, which gives us minus 4. When X is 3, that gives us 3 squared, which is 9 takeaway 12 which gives us minus three and you can see the symmetry starting to form here. Now a Nexus 4 gives us 4 squared 16 takeaway 16 which as we expect it is a 0. And for our final function, FX equals X squared minus six X, we're going to look at coming from X is one all the way up to X equals 5, X equals 1, gives US1 takeaway six, which gives us minus 5. X equals 2 gives us 2 squared. Is 4 takeaway 12 which gives us minus 8. I mean for taxes, three in three squared is 9 takeaway 18, which gives us minus 9. X is 4 + 4 squared. Take away 24. Which 16 takeaway 24 which is minus 8. I'm waiting for X is 5 in we get 25 takeaway 30, which is minus five and once again symmetry as expected. So as with our previous examples we want to draw this graphs of these functions so we can see what's going on as our value of the coefficients of X or Y is changing. So if we got F of X on our vertical axis, this time we're going from Arlo's values, minus nine. Our highest value is 6. So we're going from minus 9 - 8 - 7. 6 - 5 - 4 - 3 - 2 - 1 zero. 12345 I will just squeeze in sex and are horizontal axis. We've got one. 2. 3. Four and five letter VRX axis, and we go breakdown some minus 2 - 1 - 2. So first of all, let's look at the graph F of X equals X squared minus X. So minus two and six was our first point, so minus two and six, which is here. And we've got minus one and two which is here. We've got the origin 00. And we've got 10. 22 I'm 36 And as we expected, you can see we can join the points of here. Let's make smooth curve which will be a parabola. So this is F of X equals X squared minus X. Our second function was F of X equals X squared minus 4X. So first point was 00, which we've got. One and negative 3. Say it. We've got two and minus 4. Just hang up three and negative 3. Which is here. And four and zero, which is here and straight away. We can see a smooth curve which is a parabola. Coming through those points. And we can label up this is F of X. Equals X squared minus 4X. Lost function to look at is the function F of X equals X squared minus 6X. So our first point was one and minus five, which is here. We are two and negative 8. Yep. Three and minus 9. 4 - 8. And five and minus five, which is hit, and once again we can just draw a smooth curve through these points. To give us the parabola we wanted, this is F of X equals X squared minus 6X. So what's happening as the coefficients of X is getting bigger in absolute terms. So for instance, we can from minus one to minus four to minus six, and we can see straight away that the actual graph this parabola is moving down and to the right as the coefficients of X gets bigger in absolute terms. And that's where the coefficients of X is negative. OK, so we know what happens when we varied coefficients of X squared and we know what happens when we vary the coefficients of X and that's both of them. For quadratics, what happens when we vary the constants at the end of a quadratic well? Likewise table of values is a good way to see what's going on. So this time I'm going to use X again. I'm going to go from minus two all the way up to +2, so minus 2 - 1 zero one and two, and for my functions I'm going to use X squared plus X. X squared plus X plus one. X squared plus X +5. And X squared plus X minus 4. Now this table is particularly. Easy for us to workout compared with the other ones because we have a slight advantage in that we have a Head Start because we already know the values for X squared plus X that go here and we can just take them directly from Aurora, the table and you'll remember that they actually gave us the values 200, two and six. And in fact, we can use this line in our table to help us with all of the other lines. The second line here, which is X squared plus X plus one. His only one bigger than every value in the table before. So all we need to do. Is just add 1 answer every value from the line before, so this will be a 311, three and Seven. Likewise, for this line, when we've got X squared plus X +5. All this line is just five bigger than our first line, so we can just 7. 557 and 11. And likewise, our last line is minus four in the end with exactly the same thing before hand, so it's just take away for from every value from our first line, which gives us minus 2. Minus 4 - 4 - 2 and 2. So let's put this information on a graph now so we can actually see what's happening as we vary the constants at the end of this quadratic. So you put F of X and are vertical scale so that we need to go from minus four all over to 11 so it starts at minus 4 - 3. Minus 2 - 1. 0123456789, ten and we can just about squeeze already left it. I no longer horizontal axis. We're going from minus 2 + 2 - 1 - 2 plus one and +2. So first graph is the graph of the function F of X equals X squared plus X, so it's minus two and two. Which is a point here, minus one and 0. So first graph is the graph of the function F of X equals X squared plus X, so it's minus two and two. Which is a .8 - 1 and 0. Because the origin next 00 at the .1 two and we got the .26. For our next graph, actually sorry, now we should. Draw a smooth curve for this one first. And Labor Lux F of X equals X squared plus X. For an X graph which is F of X equals X squared plus X plus one, we need to put the following points minus two and three. So minus one and one you can see straight away that all of these points are just want above what we had before zero and one. One and three. Two and Seven. So you can imagine our next parabola that we draw through these points. Is exactly the same but one above previous, that's F of X equals. X squared plus X plus one. So what about F of X equals X squared plus X +5? Well, let's have a look minus two and Seven. Gives us a point up here. Minus one and five. Zero and five. One and Seven. And two and 11. So we draw a smooth curve through these points. See, that gives us a parabola and this is F of X equals X squared plus X +5. So finally we look at the function F of X equals X squared plus X minus four. So we've got minus two and minus 2. Minus 1 - 4. 0 - 4. One 1 - 2. Two and two. I once again we can draw a smooth curve. Through these points and this is F of X equals X squared plus X minus 4. So we can see that what's actually happening here is from our original graph of F of X equals X squared plus X, and if you like, you could have put a plus zero there. And when we added one, it's moved one up when we added five, it's moved 5 or. And when we took away four, it moved 4 down. So it's quite clear to see the effect that the constant has on our parabola. When looking at the graph of a function, a turning point is the point on the curve where the gradient changes from negative to positive or from positive to negative. And we're thinking about polynomials. Are polynomial of degree an has at most an minus one turning points. So for example, a quadratic of degree 2. Can only have one turning points. And if we draw just a sketch of a quadratic, you can see this point here would be at one turning point. We think about cubics, obviously cubic as a polynomial of degree 3, so that can have at most two turning points, which is why the general shape of a cubic looks something like this. We see the two turning points are here and here. But as I said, it can have at most 2. There's no reason it has to have two, and a good example is this of this is if you look at F of X equals X cubed. And in fact, that would look if I do a sketch over here. F of X&X that would come from the bottom left. Through zero and come up here like this. That's F of X equals X cubed, and so we can see that the function F of X equals X cubed does not have a turning point. Another example, if we were looking at a quartic curve I a polynomial of degree four, we know can have up to at most three turning points, which is why the general shape of a quartic tends to be something like this. With the three, turning points are here, here and here. Now let's suppose I had the function F of X equals. X minus a multiplied by X minus B. Now to find the roots of this function, I want to know the value of X when F of X equals 0. So when F of X equals 0, we have 0 equals X minus a multiplied by X minus B. So either X minus a equals 0. Or X minus B equals 0, so the roots must be X equals A. Or X equals B. Now we can use the converse of this and say that if we know the roots are A&B, then the function must be F of X equals X minus a times by X minus B or a multiple of that, and that multiple could be a constant. Or it could be in fact any polynomial we choose. So for example, if we knew that the roots were three and negative. 2. We would say that F of X would be X minus 3 multiplied by X +2 or multiple so that multiple could be three could be 5, or it could be any polynomial. OK, another example. If my roots were one 2, three and four. Then my function would have to be F of X equals X minus One X minus 2. X minus three. At times by X minus four, or it could be a multiple of that, and as I keep saying that multiple could be any polynomial. Right so Lastly, I'd like to think about the function F of X. Equals X minus two all squared. Now if we try to find the roots of this function I when F of X equals 0, look what happens. We had zero equals and I'll just rewrite this side as X minus two times by X minus 2. Which means either X minus two must be 0. Or X minus 2 equals 0 the same thing. So X equals 2. Or X equals 2. So there are two solutions and two routes at the value X equals 2 as what we call a repeated root. OK, another example. Let's suppose we have F of X equals X minus 2 cubed. Multiply by X +4 to the power 4. Now, as before, we want to find out what the value of X has to be to make F of X equal to 0. And we can see that if we put X equals 2 in here, we will actually get zero in this bracket. So X equals 2 is one route and actually there are three of those because it's cubed. So this has a repeated route, three of them, so three repeated roots. So three roots for X equals 2. But also we can see that if X is minus four, then this bracket will be equal to 0, so X equals minus four is a second route and there are four of them there. So there are four repeated roots there. Now what we say is that. If a route. Has an odd number of repeated roots. For instance, this one's got 3 routes, then it has an odd multiplicity. If a root, for instance X equals minus four, has an even number of roots, then it hasn't even multiplicity. Why are we interested in multiplicity at all? Well, for this reason. If the multiplicities odd, then that means the graph actually crosses the X axis at the roots. If the multiplicity is even, then it means that the graph just touches the X axis and this is very useful tool when sketching functions. For example, if we had. F of X equals. X minus 3 squared multiplied by. X plus one to the power 5. By X minus 2 cubed times by X +2 to the power 4. Now, first sight, this might look very complicated, but in fact we can identify the four roots straight away. The first One X equals 3 will make this brackets equal to 0, so that's our first roots. X equals 3 and we can see straight away there are. Two repeated roots there, which means that this has an even multiplicity. Some even multiplicity. Our second route. He is going to be X equals minus one because that will make this bracket equal to 0. So X equals minus one. I'm not actually five of those repeated root cause. The power of the bracket is 5, which means that this is an odd multiplicity. We look at the next One X is 2 will give us a zero in this bracket. So X equals 2 is a root there and there are three of them, which means is an odd multiplicity. And finally, our last bracket X +2 here. If we put X equals minus two in there. That will give us zero in this bracket and there are four of those repeated roots, which means it isn't even. Multiplicity. So as I said before, we can use this to help us plot a graph or helper sketch the graph. So for instance, and even multiplicity here would mean that at the roots X equals 3. This curve would just touch the access at the roots, X equals minus one. It would cross the Axis and at the River X equals 2 is an odd multiplicity, so it will cross the axis. And at the roots for X equals minus two is an even more simplicity, so it would just touch the axis. So let's have a look at a function where we can actually sketch this now. So if we look at the function F of X. Equals X minus two all squared multiplied by X plus one. We can identify the two routes immediately. X equals 2 will make this bracket zero and we can see that this hasn't even multiplicity. So even multiplicity. And X equals negative one would make this make this bracket 0. So this will be an odd multiplicity. Because the power of this bracket is actually one. Now we talked before about the general shape of a cubic, which we knew would look something like this. So how can we combine this information and this information to help us sketch the graph of this function? Well, let's first of all draw axes. Graph of X here. Convert sleep X going horizontally. Now we know the two routes ones X equals 2 to the X equals negative one. Now about the multiplicity. This tells us that the even multiplicity is that X equals 2, so that means it just touches the curve X equals 2. But it crosses the curve X equals minus one. I should add here that because we've got a positive value for our coefficients of X cubed, if we multiply this out, we're definitely going to get curve of this shape. So we can see it's going to cross through minus one. It's going to come down. And it's just going to touch the access at minus 2. So this is a sketch of the curve F of X equals X minus two all squared multiplied by X plus one. We do not know precisely where this point is, but we do know that it lies somewhere in this region.