Polynomial Roots and Coefficients (1 of 5: Relationship between roots and coefficients of cubics)

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“Is the heading. I m going to ask you to put a secondary heading on on this because up this is a very functional way of describing what we re to looking at today. But um traditionally historically this is not at all what it s become good boy ah. These are called and i m not i don t you could try pronouncing this one because i have it unlike um.

I like the barbarous theorem. I haven t done the research to go that i m going to be saying this right. So. This is a french mathematician.

Who worked out these results that i m going to show you um. And that s what like this is a really beautiful result that we re going to look at okay. It s all about the relationship between the roots of a polynomial and the coefficients that has now you already know what some of these are let s just rehearse for a quadratic right. We can stay any any quadratic polynomial.

As like with general coefficients in general form right. So. If i have a x squared. Plus.

Bx plus c. Equals zero. Okay out of this i can make a statement. If i say the roots are alfred beat up because it s a quadratic so it should have three roots.

I can make a statement about what happens. When i add those roots and what happens. When i multiply those winds right now if you recall how we got this relationship. What we said was look if i ve got a quadratic like this and you don t need to learn less this i m just gonna do a quick.

The other side and we re going to do it for the cubic. It according. If you have a go at something like meters and you say look this quadratic cannot have two roots right or to have two roots. I should be able to say that i could rewrite this not say it s equal to something.

But actually say it s identical to a quadratic that looks like this oops love. It okay. It s got two roots alpha. And beta.

And i ca. N t just. Say. X.

Minus. Alpha..

X. Minus. Beta. And say that s the factorization.

Because. I don t know whether it s monocore not a could be 1 or 2 or negative a hundred or anything okay. So that s what s at the front. Ok.

Now what i would do here just like we did with in so many places right because we re approaching the one object from two points of view. I d expand the right hand side and then i have some x squared terms. Some x s and a constant just like i have here and since these things are identical. I will equate them i ve compared coefficients and then i get a result out now this is for us.

What is that result. What is the sum of the roots of a quadratic. My spirit very good right there s the summer fruits and in the same way once you reverse that process you get the product of the roots alpha times beta. Which is just cnn.

Right so. A couple of things have happened number 1. The coefficient has changed and also it s gone from negative to positive thank you ok so the end product of roots now as it turns out southern products are actually not a very appropriate name. As you will see as we now generalize this result ok so now rather than just a quadratic.

We re going to have a look at a cubic at this time. We are going to fully rehearse this result rather than just say hey do you remember this we re actually going to go through the lines. Okay. So i m going to say we start with a general cubic right again.

I m going to name. They. The leading coefficient. A.

So ax. Cubed. Plus. Bx.

Squared. Plus. Cx. Plus.

D. Okay..

Now. What i want to know is something about the roots of this cubic. And how many roots should have it should have three just like a quadratic has two there should be three roots and i want to know how they relate to each of these coefficients okay. So my method is going to be just like over there.

I m going to say look this thing. Here it should be equivalent to something. Which has three roots. I want it completely factorized out right so i have that out the front because it s not necessarily money and then if there are three roots.

I m going to call them alpha beta and gamma okay in case. You re wondering gamma. I ll draw a nice pink. Eye over.

Here. This is the way i draw. Gabor s. Okay um.

Kind of like a how would you describe. It s like a ribbon. You know there s rivets that you buy their fundraising ribbons. It s like a ribbon winner with a tailor okay for a hat whoever it is okay so here s my factorization now.

I m gonna take this i m just going to expand a lot okay now i m going to give you a second to excuse me you re going to get a whole bunch of turns out of this okay get all the terms out just one by one if i take you a couple of lines. But the important thing that what you get is can you gather the x cubed terms can you gather the x squared terms can you gather the etc. Now just to make things one step easier for you before you begin your expansion. Okay before i expect you see everything here on the right is going to have an a on it right like every single thing is going to have an edge.

Because there s an a at the front okay so rather than do that i m just going to take the a out of play right. I m going to divide both sides by a it s just the null. But it s an odd zero not because it s a cubic. If a was 0.

I have a quadratic okay so find about everything i should get x. Cubed. Plus. This number of x squared s plus.

This number of x s plus. This constant okay so i m divide that all by a which just leaves me with this which is very easier to expand. But it just worked me somebody as flying around okay. So now you ve got the right hand side go ahead and expand.

It and collect the life terms. Okay i figure..

I m one step short okay you can see what i ve done is i ve actually just expand everything out. But i ve put them into groups. I just haven t quite factorized here right oh. I pointed this out just so that you know what are the tricky.

Things here is just making sure you ve got everything just make sure you have missed any terms. You re going to have three x squared terms. Now just before i you know actually factorize this how do i know i had to have three x squared terms. Like i knew i had to have three of them okay have a look at how you get an x squared.

Out of this right in what way could i get x. Squared. And the answer is you have to take these two which leaves you with don t pick an x here. Because then you won t get neck squared.

You ll get an execute right you re going to take these two are you going to take these two and then you ve got to take those two in other words out of three objects. You were trying to choose two of them right in other words picking back to your binomial that you live way at the side of here there were three choose two of those right. Which if you want to go back to of your path schools fraggle right. 3 choose.

2 is going to be able basically choose look 3 choose 1 3. Choose. 2 okay so that s how i do then we re going to be three and your calculator can check that okay that ll be more useful we go for it a bit so i also do there were going to be three. These okay you can see the 1 3.

3. 1. That pascal s triangle. Expect us together.

Okay do you see it okay. So all right now ready to factorize so i m going to in order to make sure i can compare the coefficients. I need the actual coefficients just sitting at the front. I notices it executes on both sides.

But that tells me nothing about the coefficients are written how they re related to each other so i m just going to get rid of both of them do you see that it doesn t add any useful information to me. I ve got an x cubed on both sides. But whatever okay now see where the action happens right how many x squared s do i have i m going to take out that minus sign because they all have a minus on them. And i get alpha plus beta.

Plus. Gamma right. That s how many. X.

Squared s. There are in the same way..

I ve got all of these guys take out one factor of x. That s you now these guys what do you call them you cannot really call them the sum of roots because it s not alpha the speed of scalar. But you also can t call them the product of roots because it s like alberta. Okay so so what you call these is and if i write the last one here you can see there s a pattern that i can name these will be consistent is that this is the sum of the roots.

But it s the sub one at a time just one root. Then the other one in the other one this one is still a so it s something for something you can something. But they re not one at a time they re two at a time you see them they re all paired up okay. This is also a sun.

But they just happens to be only one object. There right some three at a time okay so there s someone at times. Some to the side some three oh time those are the coefficients that will actually get and i can compare with it right so therefore and here come the first of all that rather ii. Prefer stop you here come the second of your taste.

Results. Right. He takes the coefficients and compares them the first one is here i m going to put i ve done the colors you can see again. But this makes it a little more bees you ve got midas a hospitable scrubber equals beyond a so i m going to put the by the side of the other side as is traditionally done right so.

There s the summer fruits and that by the side pops on the other side. As you should expect based on this right. Interesting no matter. How many roots there are no matter what degree of polynomial is if you add up all of the roots.

You ll always get this result right. Oh. The next one is to two times. So you go alpha vena be together alpha and gamma that s just here.

And he there s no change of sign required okay. So that s a c or a and then lastly. There s some three at a time yes be on it all the time because on pascal s triangle. You re not that far off you don t that problem when we look to fall.

You ll see what happens okay let me write down this last. One for the cubic. Okay. So this is going to be again the side it switches right.

So you re getting minus t on a and these are beauties results to eviction equation. Okay so again you can see just like we observed here like it s actually the same numbers. It s the same numbers and ” ..


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