So, good morning everyone and welcome back to the course on Classics in Total Synthesis. This is the second lecture, so in the last lecture you know we briefly talked about the syllabus which we will be covering in this particular course. So now we will move to the second part of our introduction, so where you know we talk about many technical terms as well as some explanation on retrosynthesis and so on. So there are few technical terms which you should know and in this slide I will talk about some of the technical terms. Organic synthesis you know you often heard this term organic synthesis and in some places you know people write synthetic organic chemistry but please remember both are same do not get confused organic synthesis and synthetic organic chemistry both are same. Then you also hear total synthesis, semi synthesis, formal synthesis and partial synthesis. So what are they? Total synthesis is nothing but complete synthesis that means synthesis of a molecule it can be natural product or any other molecule but starting from commercially available simple starting materials it should start from whatever is commercially available and go all the way to the final target so that is called total synthesis. Then you also see semi synthesis, semi synthesis in literature and what does it mean? It means that you are making the target molecule you are also making the target molecule in this semi synthesis route but not from commercially available starting material but from advanced precursor which has almost all the elements all the atoms present in the final molecule okay I will give one example of semi synthesis and I also will give example of formal synthesis and partial synthesis. What is formal synthesis? In synthesis of a target molecule but you do not synthesize a target molecule but instead what you do is you synthesize a key intermediate which many times advanced intermediate but that has been already converted into the target molecule by somebody else okay. So you are making only a advanced key intermediate and that advanced key intermediate has been converted into the natural product or the final target molecule by other people in few steps okay so if you make that key intermediate then you can climb that it is a formal synthesis of that particular molecule. Then you also see in the literature partial synthesis so what is partial synthesis? So if you have a large molecule it is sometimes you know you do not want to report the whole synthesis you do not want to wait complete the total synthesis and then report. So instead what you want to do is you want to make a fragment of that molecule or fragments of that molecule and then report. So such synthesis of fragments small fragment of fragments of a big target molecule is called partial synthesis. So now we will move to what is semi synthesis? When you are trying to make a complex molecule there is a possibility that from another natural source the molecule which is very close to the molecule which you are going to synthesize may be available then what you can do? You can take that molecule and then do only few functional group transformation and then synthesize the molecule which you want to make it. I will give you one example so that will explain what is semi synthesis. So this molecule is a diterping called taxol or pachyl taxol. So this is a drug used for the treatment of you know ovarian and other types of cancer. When this molecule was isolated in 1963 people did not think that this will become in a big blockbuster drug. In 1994 this was approved as a drug for the treatment of ovarian and breast cancer. Afterwards lot of synthetic efforts went towards making this molecule. But the problem is when this molecule was isolated from the bark of Pacific U tree the Pacific U tree was found to be very very slow growing tree. So if you have to isolate more of this molecule then you have to cut almost all the trees but still the amount of taxol you can isolate from these trees will be very less to cure the patients. So that time it was felt that there are only two ways one can make this taxol to be available for the treatment of cancer. One is by synthesis other one is by biochemical methods. But fortunately what happened from the leaves of Pacific U tree another molecule which is very close to taxol if you look at these two molecules this is called 10 D acetyl they call it as 10 Dap which is almost close to what you have in taxol. You can see in taxol this is the core structure and 10 Dap has almost all of the core structure except that in taxol here you have acetate whereas here you have hydroxy group. So if we can isolate more of 10 Dap from the leaves of Pacific U tree then what one needs to do is you have to carry out the acetylation selectively on that hydroxyl group then you need to attach the side chain on this hydroxyl group. If you can do that then you complete the synthesis of taxol from an advanced intermediate which is also isolated from a natural product. So this process is called semi synthesis. See fortunately what happened in this particular case this 10 Dap which was isolated from the leaves of Pacific U trees could be used for making several analogs of taxol in fact some of the analogs of taxol were found to be more active than the parent compound taxol that is one advantage.  So this way you do not have to cut the Pacific U tree to make taxol what you need is you need to isolate 10 Dap from the leaves and then do some functional group transformation and you can complete the synthesis of taxol okay. The partial synthesis again as I mentioned if you have large molecule okay if you have large molecule and of course when you want to synthesize such large molecule it takes time and there will be always competition to report such synthesis. So what you can do you can you can cut this molecule into several fragments fragment A fragment B fragment C fragment D then try to make or try to synthesize these individual fragments and then report. See here in this case a randy cancer agent called discordermalide there are four fragments I have shown only two I have shown only three if you see if you can disconnect here if you can disconnect here and if you can disconnect here okay. So this is one fragment this is one fragment and this is one fragment and you can see this is the third fragment and this is the fourth fragment okay. So there are four fragments so if you report the synthesis of any of this four okay here I have written three the fourth one is on the left hand side okay. So if you report the synthesis of any of this four then you can claim that it is a partial synthesis of discordermalide partial synthesis we also have to mention the fragment C1 to C12, C12 to C14 whatever the number that you have to mention then you can claim that it is a partial synthesis of that natural product. And the third one is formal synthesis the name itself suggests it is a formally done okay you are not really done the whole total synthesis and you have done formally. So for example we will give our own example so where we have taken this material which was prepared in few steps from commercial availability material and that can be redrawn like this. So when you redraw you can see a double bond okay and a triple bond ideally situated for an intramolecular enion metathesis okay this can undergo an enion metathesis to generate to generate this diene okay in C2. Now this diene can undergo an intramolecular Diels-Aul reaction to give this tricyclic skeleton okay. So this tricyclic skeleton is already converted into a natural product called venegral by two groups okay. So what we can claim is the formal synthesis of venegral where we have made this advanced intermediate this intermediate has been already converted into natural product venegral. So we claim that this is a formal synthesis okay. So now you know what is total synthesis, what is partial synthesis, what is semi synthesis and what is formal synthesis okay. So now before we go further into this topic what are the basic requirements one should have to carry out total synthesis okay. So first of all as I mentioned yesterday the knowledge of most of the organic reactions in your fingertips is very important. The reason is when a particular reaction does not work then you should think about alternate reactions alternate pathways. So that means you should have knowledge of wide range of reactions. So then only you should be able to face failures and then succeed. So knowledge of organic reactions and other related things in organic chemistry is very important. Then creativity, so creativity is not only required for organic synthesis but for every subject but here creativity is more important for the simple reason that how you make the molecule is more important and that is directly related to the creativity. Basically one can make a molecule it does not matter you know see it may take 20 steps it may take 50 steps no problem one can work hard and then make that molecule. But if you are creative one can make the same molecule in 5 steps and compare that with 15 steps 20 steps reported in the literature then you feel your creativity really helped in reducing the number of steps okay. So creativity plays a very very important role when you talk about synthesis and this is a natural product or any pharmacologically important molecule. So even industry creativity plays a very very important role. The third component this is you know it depends but basically I prefer that artistic taste is also very important. So it is like when you read a particular journal volume okay there may be several total synthesis reported in the molecule in that particular issue. Not all the synthesis you will remember okay after going through all the synthesis only few synthesis will linger in your mind how did they do how did they think it is so good fantastic you know so that is how you feel. That is because it is not about just synthesis how they thought about it how they constructed the molecule how they constructed a very important strategic bond. So these are all you know it is like you have to have artistic taste then only you can really think about synthesis of a complex molecule in a simpler way. Then fourth component is also very important it is persistent patience and when you go to lab many times there known reaction will not work. So you have to have patience you should know why the reaction did not go so you have to analyse what did you do wrong. So patience will really in the long run give you a long rope and then you should be able to achieve the total synthesis of molecule. So persistence is one of the major ingredients requirements for a successful total synthesis person. And skill see experimental skill varies when a first year PhD student does a reaction you know a particular reaction you will come back and then tell either that reaction did not work or you will say the reaction work with 50% yield though the reported yield will be 100% okay. So the experimental skill over a period they you know they get it. So it is important that you develop the experimental skill over a period by the time you finish PhD you should have wonderful experimental skill why I am saying is at the end of PhD if you say I tried this reaction and that reaction did not work. That means your supervisor should believe that so and so has tried that reaction and that reaction did not work. That means nobody even if they try the reaction it will not work because he has such a fantastic experimental skill and still you could not fill the reaction did not work okay. So experimental skill they are very important role that can be acquired only over a period of time and it does not come naturally it takes time. And stamina most of the times in synthetic laboratory you have to stand and do the reaction and you can imagine synthetic chemists work very hard. So it is not like 8 hours laboratory work so they work much more. So they need both physical and mental stamina. Mental stamina also is very important because of the failures which they face. So they should be very strong mentally and physically and of course one should be courageous and the character yes I can do it okay whenever they fail they should get the confidence yes I will do it I will make it work okay. So these are the 7 basic requirements for a synthetic chemist to successfully carry out total synthesis okay. So why I am insisting on this 7 important criteria because unless you have all this or at least most of this it will be very difficult to carry out total synthesis okay yeah. So now we will move to why we need to do synthesis as for any subject the first question you should ask is why should you do? Any branch any discipline any subject when you have to carry out something first question you should ask why should I do okay. So again when you ask why should you do synthesis already I told you it is a great place to use your creativity if you are highly creative yes in organic chemistry synthesis is the subject you have to work okay it gives you know so much so many problems and using your creativity you can solve most of them okay creativity plays a very very important role. The second one is scientific excitement and satisfaction see imagine when you are working on a total synthesis of a natural product okay what happens nature makes this combo so that is why we call it as natural product and same thing you want to repeat that in the laboratory so that means you are trying to mimic what nature has done and trying to reproduce in the lab okay the moment you make the natural product and see the spectrum and which matches with the natural product the amount of excitement and satisfaction you get is uncomfortable nothing is you know closely nothing can come close okay so that much excitement and satisfaction you will get so this is very very important if somebody wants to do total synthesis. Then the third point is when you talk about science when you talk about synthesis the synthesis is not only science it is also an art as I mentioned there may be so many synthesis of the same molecule so many synthesis of the same molecule but still in your textbook only one synthesis will be discussed in details that is because it is not only synthesis it is beautifully constructed the way in which the author thought about the synthesis thought about the retro synthesis and assemble it art you know artistically you would have done it so that is why you know the science and art they combine a synthesis a combination of science and art okay so those who are interested in art those who are interested in science I think synthesis is the place and if you look at the history initially what happened clearly as I mentioned in the first lecture the final step of elucidation of structure of any natural product is synthesis okay they do all destruction analysis all that degradation finally the structure of the natural product was confirmed by its synthesis okay that is what it was used but later many new reagents many new catalysts many new methodologies many new tactics came into picture then how to test their reagents and catalysts okay the best way to test your reagents new the developed reagents catalysts is to apply in total synthesis of natural product okay I have made wonderful catalysts but how do I know that is a wonderful catalyst apply it in very difficult total synthesis then you can claim yes my catalyst work and my catalyst was used by so and so in the total synthesis and it was the key reaction and it was highly successful okay so that is how the testing of reagents and catalysts should be done and even after having so many reactions so many new reagents so many new catalysts some reactions won't work okay and it is very difficult to know why reactions do not work okay you can analyze you can you can still find a solution but still sometimes you do not know that is where there is a continuous need for development of new chemistry okay so when something don't work you have to keep on thinking how to solve it how to address it how to make it work so the development of new chemistry fairly goes along the synthesis of natural product okay the same person can do or somebody else can do the same thing and try to help the synthetic chemist who are interested in the total synthesis of natural product and needless to say synthesis plays a very very important role in medicinal chemistry biology and material science and all these places all these areas synthesis contributed a lot to their growth then the major advantage or I should say one of the major advantages of organic synthesis particularly total synthesis is when you make a natural product using the same strategy to make the natural product one can make several simpler analogs or complex analogs of the same natural product and why that would be useful that would be useful mainly because when you make the analogs then using structural activity relationship studies you can also remove certain functional groups or you can add some functional groups to the natural product you can see whether the functional group X is required or a Y is required okay see they are not required when you make so many compounds so many analogs you can finally you can using the structural activity relationship studies you can say X is not required Y is not required but Z is required okay that way you can simplify the natural product and study the biological properties okay and last starting from morning you start with brush and toothpaste and many things which we use in everyday's life is as a result of synthesis many times we do not realize but it is as a result of synthesis. I always compare synthetic chemists it is a combination of architect and civil engineer okay one can simply equate you know synthetic chemist as a hybrid of architect and civil engineer there are many things which are common okay for example both of them synthetic chemists as well as architect and civil engineer their job is to imagine a structure okay this is a structure I want to build okay for architect and civil engineer buildings you know they have to imagine a huge building and how they design how they construct whereas synthetic chemists they have to imagine a structure of a molecule okay small molecule big molecule medium sized molecule then how they make it okay for both of them the goal is same then when you do that your thinking should be logical okay it should be logical while designing as well as executing because when it fails then you also should logically think why it failed and think about alternate strategies okay so rational thinking and designing is very important in both cases and civil engineer architect together they construct buildings huge buildings whereas for synthetic chemists construct molecules but it is equally difficult the reason is at least architects and civil engineers they can see the molecule see the building whereas synthetic chemists they cannot see the builders see the molecule real molecule they can only see the structure you know assume this is a structure they have to make and of course you have to obey laws of nature otherwise nothing will move and needless to say both require lot of planning see planning is very very important before you start executing and when you plan lot of creativity goes in and understanding of the subjects is very important and finally execution execution is very very important you can have all three wonderful planning with high level of highest level of creativity and deep understanding of the subject but those who execute those who execute your plan if they do not do it properly the whole thing will go back okay so that is why it is very important all the four pillars of this are very very important and both synthetic chemists architect civil engineer they use geometry models maps drawings etc then only you know they can achieve whatever they want to achieve from synthetic chemist point of view also the diversity is really huge because so many natural products around 10,000 natural products are being isolated every year okay so that when you talk about 10,000 then obviously you know the diverse natural products and similarly when you see the number of buildings even in Mumbai if you see so many types of building huge diversity okay so diversity is there when you have so much diversity automatically the thinking the designing everything is different okay the creativity you know you have to be very creative to have diverse thinking and execution okay but there are few things which are not common and the last thing which is common is symmetry or lack of you see many buildings which are very symmetrical and you also see many buildings which are not symmetrical likewise you see many natural products having some symmetry but most of the natural products do not have any symmetry so both give enough challenges for synthetic chemist as well as civil engineer and architect to construct what are the major differences the major differences if you see the architecture was very nicely developed and fully developed profession about 4500 years ago okay but when you talk about synthetic chemistry or organic synthesis it started only in 19th century okay it started only in 19th century as you know the first synthesis was reported by Wohler on Uria and the second the third major difference is the scale when you talk about building you talk about buildings in meters whereas when you talk about molecule the scale is angstrom the difference is about 10 billion fold that is a huge difference when it meters versus angstrom 10 billion fold then you can see buildings isn't it even model one can see the buildings whereas for a synthetic chemist it is very difficult to see single molecule with his naked eye though there are now spectroscopy techniques one can try to see a single molecule lot of efforts have gone in the last 2 to 3 decades to see a single molecule but for most of us it is still you know a far away subject to see a single molecule unlike architect and civil engineer who can see the buildings and accordingly they can visualize so that means that much difficult for synthetic chemist to see and then make the molecule and last but not least major difference is when you construct building the buildings are meant for living when you construct a building all the buildings are for living whereas when you make molecule when you make molecules molecules are for life so these are the major differences between synthetic chemist and combination of architect and civil engineer. So here I show to important manmade architecture I had an opportunity to visit both the places a few years ago so this is a place called Tanjavur or Tanjur it is more than 1000 years old still maintained it is a UNESCO certified place and the other one I am sure all of you know Taj Mahal these are all very very impressive manmade structures there are many such manmade structures in India just I am showing only 2 places where I have gone and took these 2 pictures it is wonderful okay see this tells the creativity man's creativity how one can be creative and construct buildings okay so when you talk about symmetry you can see Taj Mahal you can see a clear symmetry okay and similarly you can see symmetry in the tower of Tanjavur temple okay huge accomplishment huge accomplishment several centuries ago several centuries ago we are much ahead of others in terms of architecture and construction okay so I will stop here for today's lecture and I will continue this introduction in the next lecture so here in to summarize in this lecture what we discussed is what are the basic requirements to become a synthetic chemist and why we need to do synthesis and how a synthetic chemist can be compared with an architect and civil engineering okay so now we will we may need one more lecture on introduction okay so we will continue our discussion in the next class and complete the introduction and we will move to natural products having three number ring okay thank you


In practice, this is not anything we're doing manually anymore. We're buying machines to do it. The largest company in the world selling that type of machine is called Illumina. This is one of their machines called Heisek X10. That's sexy. We have 10 of them at SciLifeLab. Just to give you a ballpark idea, you would need, they cost roughly one million dollars each. You have to buy 10 of them. Each of those machines are going to generate, in one run, taking roughly three days, they're going to generate 1.6 or 1.7 trillion base pairs. I'll say that again. 1.7 trillion base pairs. Compare that to the size of the human genome. 3 billion base pairs. Again, 3 billion, not the trillion. So we're talking about hundreds of humans' genomes worth of information. Now, the caveat is that we're going to need 30 or 50 times overlap to be able to say anything for certain. It takes three days to run that, but if you have 10 of these machines, the throughput we're getting is roughly equivalent to 20,000 human genomes every year. There are certainly sites that have much more than 10 of these. The problem, that's a great deal if you actually need to sequence 20,000 human genomes a year. It's cheap by some definition of cheap. The cost per human genome might be 1,000 dollars, which is amazing. But again, that assumes that you are willing to pay 20,000 times that per year. Still, this means that genome sequencing is more accessible than ever. I remember the days when we were talking about a cost of five or 10 cents per base pair. Today, it's completely ridiculous to talk about sequencing cost per base pairs. We're only talking about sequencing cost per human genome. This scale might not look as impressive as it is, but A, it's logarithmic. B, compare the green line how the sequencing cost has dropped with Moore's law indicating how fast transistors are growing. So the cost of sequencing is going down much faster than the performance of Q computers is going up, which poses pretty big challenges for how to deal with all this data. I would already say that $1,000 is so cheap that within a few years, if we don't know exactly what's wrong with us when we get into the emergency room, they're going to sequence us just in case. It's not as expensive as many other tests they do anyway. But that is the high end. There is also a low end of this. Oxford Nanopore, they have a device that it's literally like a decimeter long or so. You're connecting it to the USB port of your computer. Sure, you need a bit of sample material, and you need some reagents here. So it's per base pair, it's significantly more expensive than the luminous. But if you only want to sequence a few small genomes, say a COVID genome or something in your own lab, this is simple and accessible enough that we can do it. And they're not that expensive. We could certainly afford buy one if we wanted to. So I predict that this plot will keep going down another order of magnitude. This is so cheap that we could do it everywhere. And somewhere here, we start to change not the laws of physics, but the laws of science. Today, there is no other method anywhere in science that is generating as much information as quickly as genome sequencing. And I'm not limiting this to live science. I think this is valid for any field of science. And sure, the information is limiting. It's just letters, right? A, G, C, and T. But as you're collecting this much information, we're pretty much only limited by imagination what we can do for it with it. I'll show you one example that's not all part of this class, but just to show you crazy ideas we can use this for. And then I'll get back to the proteins in a second. So proteins, we've talked about protein structure, but there is actually a structure in DNA too. So this is just a schematic model, but your DNA, my DNA is somehow going to be curled up as strange ways in our cells. It's not completely random. But if I now want to start expressing a gene here in the cell, I'm going to need to find that start code on right and the Tata box needs to bind to the right place in the DNA. How do we find that? I'm going to need to uncurl this ball some way. Is this random or is there some sort of average systematic structure here? I would like to determine the structure of DNA, but I can't really crystallize it, because it's not that rigid and repeated. Well, what Jared Slippermaneiden and Eric Launder did about a decade ago is that they came up with a very smart method that briefly they're adding small chemical linkers so that if two parts of DNA are close to each other in space, we tie them up and then we create a small break around them. And then we use shotgun sequencing. And if I now sequence these small fragments around that particular break, I will now get a small sequence of letters here and a small sequence of letters here. And all I know that these had to be close to each other spatially, not in sequence, but in space. Now I can, of course, also map these letters if it's 25 or so to a particular position in a particular chromosome. And that means that I can now start to say what part of chromosome 14 is close to what other part of chromosome 14. So I'm literally using sequencing and sequence information to derive three-dimensional coordinate structure information of DNA. The difference here, though, is that you can get billions of points like this. So you're getting a huge amount of data, much, much, much faster than X-ray crystallography or cryo-EM. And based on that, they were able to derive fairly amazing results about globules in different shapes, whether the DNA is random or whether you have some sort of hierarchical structure. And this is beyond the class, but I need to give you the result, right? Their argument is that DNA is quite hierarchical and shown by the colors here. So that DNA, the structure of DNA is really designed in such a way to make it easy for me to pull out just one gene, decode that gene, and then let that gene relax back into the DNA. And that was a beautiful example on the cover of Nature about a decade ago from Leonard Mirny when he showed this in 2D, right? So that's the whole point is that you want to draw a curve so that things that are close in the space in the curve should also be reasonably close in sequence, but you can never do that perfectly. So remember, sequencing provides more information per cost and time than any other method in science.


We're talking about the PIB, how communities should be treated in this matter, how much they're really entitled to if they're the host community and how other benefiting communities should also be treated, how much exactly they're entitled to. And now joining me to look at this tonight is the SSC to the President and Public Affairs, Ajory Angalale. Hi Ajory, good evening. Good evening to you. Good evening to you. Thank you for joining us from Abuja. Now looking at the oil revenue sharing formula, is the signing of the PIB a sharing news at all? Well, I can see on the banner on the screen it says PIB passage. The PIB has been passed now for quite some time, we're beyond that stage. It has now been extended to Mr. President. So now we move from PIB to PIA. It is the Petroleum Industry Act and we are quite excited about the fact that we've been able to finally break this 20-year jinx and really bring our nation's oil and gas industry in total consonance with the very best. So there's really quite a lot to get into when you look at the content, I'll be excited about. Yes, are you still there, Ajory? Yes, I'm with you. Okay, okay. Now, yes, I agree with you. The PIA, the PIB, once it's passed, it's not. Yeah, I understand that. Yeah, now, 3% versus 30%. How do you want to justify this? Also, isn't this opening up the leeway for fresh agitations in the near future when host communities will begin to say, well, they're already saying that, and well, maybe we can say, let's play the devil's advocates. I mean, isn't this opening the leeway for fresh agitations in the near future? You know, by virtue of the question, this is a question I've gotten a lot lately, and unfortunately, when the question is asked, it's often asked without any kind of context or any kind of specificity whatsoever about what 3% represents and what the 30% represents. You hear rumors like, oh, they're only giving 3% to the Niger Delta development. They're giving 30% to the North. These are the kind of simplistic accusations that we've been hearing. But I want to, for our viewers, I want them to understand exactly how this works. I'm going to break it down really quickly. First of all, the 3% that we're talking about is not 3% of profit oil or 3% of revenues of the NNPC Limited, what will be the NNPC Limited over the next six months. We are talking about 3% of the total actual annual expenditure of the entire oil and gas sector in Nigeria. That is, last year, that was 16 billion US dollars, 16 billion US dollars. Do the mathematics on that, 3% of that is approximately 500 million US dollars. Now, that's based on last year's estimates of 16 billion US dollars. Okay. Now, if you recall, last year was the height of the COVID-19 global pandemic, where you had production shutdowns around the world, where you had, of course, we still have OPEC curves on our production. So instead of producing 2.3 million barrels per day, which we're capable of doing, we only were doing 1.4 million barrels per day, and we continue to do that. So the implication of that is that in the next few years, we're going to return to our normal production, meaning that the annual operating expenditure is going to rise from the 16 billion it was last year, up to 25 billion into 30 billion. When that happens, the 500 million we're talking about for the host community's development trust is going to skyrocket to nearly a billion. Already at 500 million, it's far greater in one year. The trust is going to be receiving more than what the NDDC receives, because the NDDC does not receive up to 500 million dollars a year. That's point number one. Point number two is when we talk about 30%, we're not talking about 30% of the total annual operating expenditure of the entire sector. Those are two different. It's just like comparing 3% of your salary compared to 3% of your company's total profit. You get two different figures. That's basically what is being done when people compare 3% to 30%. It's two different worlds. So what we're saying is this, the 30% is not for host community's development. It's not to go and build hospitals in the north. That is not how it works, but that is what some political opposition figures are trying to deceive Nigerians into believing. How this works is the 3% is for the development of host communities. So building whatever it is that those host communities need, schools, hospitals, bridges, whatever it is that they believe their needs are, that's where it's going to go. And they're going to be at the decision-making table, unlike NDDC, unlike derivation principle, which has largely failed in terms of Niger Delta development. Now, the 30% we're talking about is not for to develop people's communities in the north or anywhere else. It is for the development of frontier basins across the country. By the way, people don't recognize that we already have frontier basins in a number of states. A number of states, as you know, is not in the north. We have frontier basins in Cross River. Cross River is not in the north. So what we're saying is that this 30% we're talking about is not 30% of the entire sector's output, like the host community development's 3% is. It is 30% of NNPC Limited's revenue. That is not including IOCs and all of that kind of stuff. So that's point number three. Now, the final point is when we talk about frontier basin exploration, how it works is that 30% we're even talking about is going to go to the development of oil extraction infrastructure, exploration activities, and then the production infrastructure. That is to actually extract the oil from the ground. You have to put infrastructure in place to achieve that. So we're not going to be building hospitals and schools in the frontier basins. We are limiting the focus, the scope in infrastructure development in those areas so that Nigeria can expand the pie, increase our revenues, and just the same way that Niger Delta Oil Money is shared to all of the Federation, all local government areas and funds the entire country. It is the same way that any oil that is found in the north or in the east or in the west, it will still have the same revenue sharing formula. It will also go to all local government areas of the Federation. So we're really imploring all Nigerians to really reject the kind of divisive analysis out there that suggests that everything must be viewed from a religious or ethnic lens. Brilliant. Okay. Good summation. Thank you. Well, on the last note, let me take it up in this as well. 50 billion's worth of investments lost in the past before we could get the big past in the first place. Why did we have to go this long route in the first place? Yeah. Well, there's no doubt that we have had a huge political interference over the course of the last several decades. I'm not breaking news this evening on Plus TV by saying that there has been massive corruption in governance in Niger of the last several decades. And the implication of that is that people from the private sector are able to leverage on that to make excessive profits by bribing certain officials in certain places to retain their undue advantages over the sector. That has now been broken because President Mohamed Dubehari is not a leader who you can visit in the middle of the night with a briefcase full of euros to make him change his mind against what is in the interest of the nation. And as a result of his integrity, we've been able to basically come up with a document that's a truly win-win document, not just for the Nigerian government, the Nigerian people, the Nigerian economy, but also for the investors who are ready to do legitimate business in Nigeria. What will not be allowed within the PIA now, the Petroleum Industry Act, is we have put in a few measures which are very important. Number one, unprecedented transparency measures stipulating that all actors in the sector must now undergo a forensic technical audit conducted by an independent auditor of their books every single year. Everybody's books will be open once a year every single year for public scrutiny. That is not just the international oil companies, that also involves even the regulators, the new regulators that will be established by this act, the Nigerian Upstream Regulatory Commission, the Nigerian Midstream and Downstream Petroleum Regulatory Agencies, they will by this act be mandated to open up their books to independent auditors annually for public scrutiny. That has never happened in the history of the country. That is happening now. Secondly, the President put in place new tax reforms and regulations in the sector. For the first time, we now have a comprehensive set of financial sanctions which we will levy out to any stakeholder in the sector who is involved in tax avoidance, tax defaulting, who is involved in non-remittance or under-remittance of royalties and monies owed to the Federation, who is involved in erroneous accounting practices internally, et cetera. These are the measures that will transform the oil and gas sector for the better and Mr. President has really done well for the country in ensuring that this finally gets done. Thank you, Adjuri. Thank you very much. I'm going to have to get back to you in coming days to look at this issue and other some issues in the news. Thank you, Adjuri. You make it very difficult for me to stop your conversation. Thank you, my brother. God bless you and have a good night. Okay, you too. Bye. Cheers. Hello. Hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


Well, class, welcome to episode 12. This episode is on the fundamental theorem of algebra, and it's sort of a continuation of what we saw in episode 11. I'm Dennis Allison, and I teach mathematics in the math department at UVSC. I'd like to begin today with our objectives. We ought to actually should go to the objectives. We want to look at the fundamental theorem of algebra, and then we'll look at complex roots of a polynomial, complex number of roots. Then we'll look at the so-called complete factorization of a polynomial. And finally, we'll look at a cubic formula. So I'd like to begin today with an example to sort of tie this episode in with the last episode. Let's take a look at this polynomial that I have on the green screen. I'd like to factor the polynomial, negative x cubed plus 7x squared minus 15x plus 9. And this is very similar to some problems we worked in the last episode. If you remember, we use a rule and we use a theorem. The rule is Descartes' rule of signs. And how does that go? Who can remind us? Descartes' rule of signs. Steven? The number of sign changes in the polynomial equals the number of positive roots. Right. So we have a sign change here. We have a sign change here. And we have a sign change there. We go from negative to positive, positive to negative, and then negative to positive. So three sign changes, what does that tell me about the roots of this polynomial? There are either three or one positive roots. Right. So over here on the side, I'll say for positive roots, there are either three or one. And of course, we're referring to real number roots. When we say positive or negative, those are real numbers we're talking about. And how do I find out the possible number of negative roots? Substitute negative x in for x. Right. Substitute in negative x for x. So if I substitute in a negative x, we'll count the sign changes after this. So this will be minus negative x quantity cubed plus 7 times negative x squared minus 15 times negative x plus 9. And when we reduce this, this will be a positive x cubed and a positive 7x squared and a positive 15x and a positive 9. How many sign changes do we have there? Zero. We have no sign changes. And therefore, there are no negative roots. Now, that's particularly helpful because now I know I don't need to check any of the negative rational roots that we're going to come up with in just a moment. So up here, for negative roots, I'll say there are none because there are no sign changes. Are there any questions about that? Does everyone agree? Yeah. On your positive roots, how did you come up with it? See, they're going to be three or one. Three or one? Well, what we do is we count the sign changes. There were three sign changes up here. And then we take either that number or we reduce it by two. So either three or one. And we can't reduce it by two more would go into a negative number of positive roots. You have to have at least zero or larger positive roots. Why did you choose to reduce it by two? Because the positive roots come in pairs. So if you remove one positive root, you have to take them out two at a time. OK, now that we know, now that we have an idea anyway of what the roots look like, we go to the rational root theorem, which says that we look at the divisors of 9 and we call that little p. And we look at the divisors of the coefficient negative 1 on the highest term. We call those divisors q. And then I'll look at the ratio p over q. And I'll put the p over q over here on the side out of the way. Well, p has to be a divisor of 9. That would be plus or minus 1, plus or minus 3, or plus or minus 9. And what would be the choices for q? What are the divisors of negative 1? Negative 1? Well, plus or minus 1, actually. So because either 1 or negative 1, either 1 divides those. We're talking about integer divisors, not fractional divisors. So if I take my choices p over q, I could have plus or minus 1. That's where I take plus or minus 1 over a plus or minus 1. And then p over q, 3 over 1. That's plus or minus 3. And then plus or minus 9. But from what we just learned up here about the roots using Descartes's rule of signs, there are no negative roots. So I know that I can actually reduce this list to only the positive choices, 1, 3, and 9. So I really only have three numbers to check. And if there is a rational number, that is a fraction or an integer, that will make this polynomial 0, it has to be one of these three integers. Any other root would have to be an irrational number or maybe a complex number. Well, let's check these one at a time. I want to find out if 1 is a root of this polynomial. And that's equivalent to saying, let me write that question down, is 1 a root or a 0 of p of x? And there's an equivalent question I could ask. And that is, is x minus 1 a factor of p of x? And the reason I know these two statements would be equivalent to each other is by the factor theorem from a couple episodes back. So to find out if x minus 1 is a factor or not, I'm going to divide by x minus 1 into the polynomial. And if I divide by x minus 1, I'll use synthetic division and I put these coefficients inside. And I'm going to divide by x minus 1 outside. But you remember if I throw away the x and call it a negative 1, and then if I change the sign to a plus 1, I can actually add out here. So we're just sort of reviewing some of the concepts we've seen earlier. So it ends up that I'm actually going to be dividing by the very same number, plus 1.  And then I multiply. 1 times 6 is 6. And I add and I get negative 9. And then I multiply 1 times negative 9 is 0. So what that tells me is x minus 1 is a factor because I got a remainder of 0. And in fact, p of x can now be factored as, let's see, it's going to be x minus 1 times what polynomial? Negative 1x squared. Negative x squared, OK. Plus 6x. Plus 6x. Minus 9. Minus 9, very good. Now let me just erase this synthetic division and I'll move this back up above there so we can keep the factorization together.  And in order to factor this, I think what we should do is maybe factor out the negative 1 to make that a positive x squared term. So negative the quantity x minus 1 times x squared minus 6x plus 9. And now this trinomial factors again. Perfect squares. It's a perfect square. Yes, how does that factor? X minus 3 squared. X minus 3 squared. And so all together I would say that the roots of this polynomial are 1, 3, and then 3 again. I'll list it twice because it came from two factors. And you see those were two of the numbers over here that I was going to check out of the three. And this is the factorization of the polynomial. So p of x factors into negative 1 times x minus 1 times x minus 3 squared. And I can use this information to now draw the graph. So let's just put the graph over here on the side. And if I want to graph this polynomial, I'll go over to 3 and I'll put a dot there. And I'll put a 1 right there. And who can tell me what the graph will look like? Should I come in from above toward 3 or should I come in from below? From below. From below, exactly. Stephen, why should I come in from underneath? Because the x cubed term has a negative on it. Yeah, you see this was a negative x cube. And so that tells me that this graph has been inverted. And rather than going up on the right, it goes down on the right. So my graph is going to come down like this. And when I come to 3, what's going to happen? Will I cross at 3 or will I turn and come back? Turn and come back. Turn and come back because this has multiplicity 2, which means this looks something like a parabola right here at 3. It's going to turn. It's rounded right there. It comes back down. We don't really know exactly how far down it would to go. We're just making a rough sketch. But eventually, I'll have to turn and come back toward 1. And at 1, what's going to happen? Will it pass through or will it turn around? It'll pass through. It'll pass through because this has multiplicity 1. So it passes through and this graph now goes up on the left-hand side. Could anyone tell me what the y-intercept is right here where this function crosses the y-axis? 9. It's 9. Yeah, Susan, how did you know that? Plug 0 in for x from the original function. Exactly. The way you find y-intercepts is you let x be 0. And if I plug in 0 for x, Susan just did that in her head. She got 0 plus 0 minus 0 plus 9 is 9. So this is actually 9 on the y-axis. And you can see that my scale obviously must be different on the y-axis than it is on the x-axis. And I wouldn't guarantee that there's any accuracy in how low this point is right here. It could actually go lower or maybe not quite so low. But this is the general shape of the polynomial function p. So I'll just put a big p beside it to say that's the graph of p. So you see what we're doing is we're taking polynomials that are of larger degree than we're used to factoring. And we're using Descartes rule of signs and the rational root theorem to find factorizations to get the roots. And the roots are the x-intercepts. I plot the x-intercepts and I use multiplicity so that I can draw their graphs. Now, you know, in the process of doing problems like this, I think it's easy to lose track of some real shortcuts that we may overlook in the middle of all these theorems and rules. So let me show you a couple problems that have shortcuts that we might otherwise overlook if we were only thinking about the rational root theorem and the rule of signs. Take, for example, this polynomial. I think I'll call this one f of x. And let's say this is x to the fourth minus 5x squared plus 4. Now, I'm thinking rather than counting the number of possible positive and negative roots this may have, I think we can actually factor this by sight because this is in a quadratic form. And you see, if I think of x squared as my basic term, this is the square of x squared. So I could put an x squared here and an x squared here. And I don't think we need the rational root theorem to factor this. Does anyone see how to complete that factorization? Minus 4 and minus 1. Minus 4 and minus 1. Yeah, so rather than looking for sign changes and the choices for little p and little q, let's just factor this thing if we can do it. And x squared minus 4 factors as the difference of two squares in the x plus 2 and x minus 2. And x squared minus 1 factors in the x plus 1 and x minus 1. And therefore, what are the roots of this polynomial? Plus and minus 2 and plus and minus 1. Right, plus or minus 2, plus or minus 1. Those are exactly the values we would have come up with if we had used the rational root theorem. But this is a lot quicker. This is a lot quicker, absolutely. So we don't want to forget about the easy ways to factor polynomials when we're thinking about these other methods. Let's go ahead and graph this while we're here, by the way. I'm going to put an x intercept at 2 and at 1 and at negative 1 and at negative 2. And because this is a positive x to the fourth, I know my graph comes in from up above. And because each one of these roots has multiplicity 1, every one of those factors is different. I know that every time this function passes through, it's intercept. And it looks like this. This is the graph of F. I would guess that the graph has a higher maximum between negative 1 and 1 than it does have a minimum between 1 and 2 or negative 2 and negative 1. Why would you guess that point would be higher than these are low? Because the space in between it is bigger than that. Yeah, just sort of from a layman's point of view, because the space is wider, we have more time to wander away before we come back. So I would guess that it will go higher there than it goes down here. Can anyone tell me what that y-intercept is right there? Four. It's four. Yeah, it's the constant term on the end. Because if you let x be 0, you get four. Looking back at this function, I noticed this is an even power, x to the fourth. This is an even power, x squared. And this is an even power, because we could think of it as being x to the 0 power. That's an even power. And when you add or subtract multiples of even powers of x, what kind of a function do you get over here? An even. You get an even function. Yeah, and it's been a while since we talked about even functions. But an even function is symmetric about the y-axis. And you notice, if you were to fold this graph across the y-axis, it would be perfectly symmetrical. And so as soon as I see the graph on this side, I know the graph is just the mirror image on the other side. So using properties of symmetry, it helps me to see what this graph would look like. OK, one more example. This is another one that I think we could actually factor without going to the rational root theorem. And I'll call this g of x. And this is the function 2x cubed minus 3x squared minus 2x plus 3. Now, if we were not in our toes, we'd probably say, let's count the sign changes using Descartes' rule of signs and see what would be the possible number of positive and negative real roots, then look for the rational roots, but using the rational root theorem. But I think we can actually factor this. Does anyone see a way to factor this? Grouping. Factor you by grouping, exactly. So if we group these two terms together and group these two terms together, that's including the negative, so I'll make that a little bit wider. In the first pair of terms, what can I factor out? x. More than x. 2x. 2x. Not 2x. x squared. Well, an x and another x would be an x squared. Yeah, so it's x squared times 2x minus 3. And what can I factor out of the second pair? Negative 1. A negative 1. And then that gives me a 2x minus 3. You see, my goal is to try to produce the same factor in both terms there. And now I do have a common factor, so factoring by grouping allows me to express this as 2x minus 3 times what? x squared minus 1. x squared minus 1, right. And then we can factor that further.  And we got our three roots, but we didn't have to appeal to the rational root theorem, because this problem was a rather rare one where we could factor by grouping. In this case, the roots are, let's see, what would be the root for this one? 3 halfs. Yeah, you see, the way you find the root, technically, is you set it equal to 0 and you solve for x. And x is equal to 3 halves. So the number that makes this 0 is 3 halves, which means this graph has an x-intercept at 3 halves. And then the roots for these two factors are negative 1 and plus 1, so I'll just say plus or minus 1. And then if I draw this graph, I think we can squeeze it in over here, I'm going to locate plus 1, minus 1, and plus 3 halves. Now that's got to kind of squeeze that in between the 1 and the 2. And this was a positive x cubed. Therefore, my graph is going to be going up on the right-hand side. And each root has multiplicity 1. So it's going to pass through each x-intercept. And this is the general shape of the graph. The highest point is, well, I should say the y-intercept is at 3, but I don't really know that that's the highest point. In fact, I wouldn't be surprised if this graph doesn't actually look more like this. So that the highest point is actually not quite on the y-axis, but maybe skewed to one side or the other. You could check this on a graphing calculator to find out. But we wouldn't expect that there would be perfect symmetry between the negative 1 and the plus 1. So it may actually have its maximum value just off the y-axis. Now, on an exam, when you drew this graph, if you put its maximum right at 3, I wouldn't count off for that. Because without further investigation, you're just trying to sketch a rough sketch of the graph. And that would be sufficient for me. Now, in all the examples that I've been considering, the coefficients that I've been putting on my variables have been integers, like a 2, a negative 3, a negative 2, and a 3. What if I had put fractions as a coefficient? Suppose we had something like this. f of x equals 1 half x cubed plus 1 third x squared plus 5 sixth x plus 1. Now, of course, this makes the problem look much more complicated, but I can get it down to integers if I factor out the fraction with the least common denominator, namely 1 sixth. If I factor out a 1 sixth, this would be 3 x cubed, because 3 sixth is a half. What would the next term be? 2 x squared. 2 x squared. And the next term would be? 5 x. 5 x. And the last term would be? 6. So if I wanted to graph this polynomial with the fractional coefficients, what I would do would be to graph this polynomial, just as I have been graphing the others. And then what does the 1 sixth do to the graph? Does it squash it? It's going to sort of squash it. Yeah, so when I draw it, I won't make it quite so elevated and wouldn't make the valleys quite so low. It would be a little bit calmer near the x-axis. So if you put fractions in here, it's not really a problem in going about graphing it. But now let's consider yet another possibility. What if I have a polynomial function, but this time I use a complex number as a coefficient? Suppose we have x squared plus ix plus 2. Now here's what's different about this problem is I have a complex number, in this case an imaginary number, as a coefficient for the polynomial. Now I wouldn't be able to graph this on the x-y plane because I can only graph real numbers on the x-y plane. And I have complex numbers introduced here. But if I wanted to find the roots of this polynomial, I could use the quadratic formula. Using the quadratic formula, let's see, x equals negative b plus or minus the square root of b squared minus 4ac all over 2a. What is the value of a in this problem? 1. What's the value of b? i. i, and the value of c is 2. And if I substitute those in, I get negative i plus or minus the square root of i squared minus 4 times 1 times 2. All over 2a, I left off an a there. All over 2a, which is 2 times 1. If I reduce this radical, what is i squared? Negative 1. And here I have negative 8. So we have negative 1 minus 8 all over 2. And if I simplify that, I have negative i plus or minus the square root of negative 9 over 2. What are we going to do with the square root of negative 9? Pull out an i. We can pull out i. Yeah, you see, we could write this as minus i plus or minus the square root of 9 times the square root of negative 1 over 2. I'm just separating the negative 1 from the 9. And how much is that when I reduce it? 3i. 3i. So we have negative i plus or minus 3i over 2. Does everyone agree with that? Now I actually have two answers there. One answer, one root, I'll call it root number 1 or x number 1 is negative i plus 3i over 2, which reduces to how much? i. i. And another root, I'll call it x number 2, is negative i minus 3i over 2, which is how much? Negative 2i. Negative 2i. So you see, I do have two roots for this quadratic function. But both of the roots in this case turn out to be imaginary numbers. And if I were to substitute either one of those numbers in to my original equation back here, either one of those should satisfy it. Why don't we just try checking the i just to see that it does? And the purpose of this example is to show you that the polynomial functions that we could deal with are now even larger, it's an even larger class of functions, because we can now use complex and imaginary coefficients. Is there an imaginary graph that you could? There isn't on the x, y coordinate plane because, well, in this case, we have two roots, i and negative 2i. And there's no place to locate them on the real number line on the x-axis. You could locate 1 and negative 2, but not 1i and negative 2i. So there's no coordinate plane that maybe wouldn't be x, y that you could graph? Well, as a matter of fact, in a course called complex variables, there is a way that you can do something like this, but we won't get into that in this course. So right now, all we can do is just find the roots of the polynomial without actually drawing a graph of it.  Now, does that make it 0? Let's see. Well, what is i squared? Negative 1. Negative 1. And here's another i squared. Negative 1 plus 2 is 0. Yeah, so that's exactly what we were expecting to happen. And if you substitute in a negative 2i, you'll get 0 also. So those are the two roots. So if I were to put complex coefficients, complex or imaginary coefficients on the variables, I may get complex or imaginary roots for the problem. Well, this leads us to a theorem that is on the next graphic. Let's bring up on the screen. And this is called the Fundamental Theorem of Algebra. This says that every polynomial of degree n, 1, or larger, with complex coefficients, sort of like what we just saw a moment ago, has at least one root. And that root may be a complex or an imaginary number, just as it could be a real number. Now, there's a generalization of this, which is in the next graphic that's called the Complete Factorization Theorem. And it says every polynomial p of x of degree n, 1, or larger with complex coefficients has n roots, which we might call c1, c2, c3, up to c sub n. So that p of x is equal to a times x minus c1 times x minus c2 times x minus c3 out to the factor x minus c sub n for some constant a. Now, let's just discuss those two theorems for just a moment. Suppose I have a polynomial with complex coefficients. Let's say my function is 1 plus i times x squared minus 3i times x minus 1 minus i. This would be a quadratic function with complex coefficients. This is certainly complex. This is complex. Negative 3i is considered a complex number because it's actually 0 minus 3i. So it has a real and an imaginary component. But it's also just an imaginary number. And according to the theorem, the fundamental theorem of algebra, this has at least one complex root. Well, let's see what that complex root would be. If I solve for x, I'll get negative b, which is the negative of negative 3i, plus or minus the square root of b squared, that would be negative 3i squared, minus 4 times a times c. Now, c has a negative in there, so that's going to be negative the quantity 1 minus i. All over 2a, so that's all over 2 times 1 plus i. If I reduce that, my first expression is just a 3i, plus or minus the square root. What would be the square of negative 3i? Negative 9. OK, I'm going to write that as 9i squared, and then I'll make it negative 9 on the next step. 9i squared, there's a negative in the parentheses here.  And this is all over 2 times 1 plus i. Now, if I reduce what's under the radical, I think it's going to be quite a bit more compact. 3i plus or minus the square root of, this is negative 9, right here, negative 9, plus 4 times. Now, how do you multiply 1 plus i and 1 minus i? I'm thinking that's the difference of two squares. So when you multiply, what do you get? You'll get negative 2, won't you? You get 1 minus i squared.  And I'll come back and reduce that in a moment. Over 2 times 1 plus i. And that will be 3i plus or minus the square root of negative 9 plus 4 times 2. I think that'll be a 2 right there. 1 minus a negative 1 makes it a 2. Over 2 times 1 plus i. And this is 3i plus or minus the square root of negative 1 over 2 times 1 plus i. Well, of course, the square root of negative 1 is i. So this is 3i plus or minus i over 2 times 1 plus i. So this is sort of an exercise in patience in working out all this algebra. So I actually have two roots. So let's separate the two roots. One of the roots I'll call x1 is 3i plus i over 2 times the quantity 1 plus i. That's going to be 4i over 2 times 1 plus i. So let's just make that 2i over 1 plus i. What I've done is to cancel off a 2 on top and bottom. The other root is x2 equals 3i minus i over 2 times 1 plus i. And if I cancel off a 2 on top and bottom, what will this be? Just 1 over 1 plus i. Just i over 1 plus i. Now, a person might argue, well, Dennis, this is not a complex number. This is a quotient of two complex numbers. So what I'd like to do is now convert these both into complex numbers as such. You remember the fundamental theorem said if you have a polynomial with complex coefficients, then you will have at least one complex root. I think these are both complex roots. So let's see what the complex roots are.  And let's find out what root, what complex number this is. x1 is 2i over 1 plus i. Now, if I want to make this into a standard form complex number, what I do is multiply by the conjugate on top and bottom. And the conjugate is 1 minus i over 1 minus i. So I'm putting the conjugate on top and bottom. And what I do is just change the middle sign there in front of the i term. So when I multiply on top, I have 2i minus 2i squared. And on the bottom, I have the difference of two squares, 1 minus i squared. And if I reduce the denominator, what will that be? That'll be a 2. And if I reduce the numerator, how much is negative 2i squared? 2? No, 4. Actually, it's a plus 2 because it's negative 2 times negative 1 is plus 2. I'm going to put that in front. And then add the 2i on the back. So I'm just moving this guy over to the back side. That way, I have my real and my imaginary component in what we call standard ordering. So if I divide by 2, I get 1 plus i. So the number that I had down here that I called x1 as this ratio is actually just 1 plus i. And what is x2? Well, if I reduce i over 1 plus i, what I'll do is here is multiply by the conjugate on the top and bottom, just like I did up above. And the numerator is going to be i minus i squared over 2. 1 plus i times 1 minus i is 1 minus i squared or 2. And that's going to be, let's see, that'll be a plus 1 plus i over 2. And that's the complex number 1 half plus 1 half i. So the original function has two complex roots.  Now the fundamental theorem of algebra said there would be at least one complex root. But then in the second theorem called the complete factorization theorem, it says you will have as many complex roots as you have the degree on the polynomial. This is the second degree polynomial, so I get two roots. Sometimes I get the same root twice, and then I still count that as two roots. They just happen to be a multiplicity of multiplicity, too. So it is possible that you can have complex roots for a polynomial that has real or complex coefficients. And of course, I think a very reasonable question is, who would ever use this information?  Well, if you remember, in the first days of this course, we said that this is a surface course for many different disciplines. And those students who are going into engineering, physics, mathematics, and let's say computer science, computer engineering, these students will be using this information. If your major is history or art or psychology, this may not be of interest to you. But there'll be other things that are of interest for those disciplines. So we're just covering a number of bases for the many different disciplines that require their students to take this course. Now, let's see. Suppose I'm looking for a polynomial, p of x. And this polynomial is going to be of degree 3. In other words, it'll be a cubic polynomial. And suppose I told you that its roots, or its zeros, are 1, 1, and negative 4. Can anyone tell me a polynomial of degree 3 that has these roots? Steven? x minus 1 times x minus 1. OK, I'll just put a square on that. Times x plus 4. Times x plus 4, exactly. So if I go back and build the factors, x minus 1, x minus 1 again, and then x minus negative 4, or x plus 4, this is a polynomial that has those roots.  So then if I combine terms, this is x cubed plus 2x squared minus 7x plus 4. Here is a polynomial of degree 3 that has these three roots. Can anyone think of another polynomial of degree 3 that has these same roots? There are actually many others. You could multiply the whole thing by 2, or by a coefficient? Yeah, multiply by 2, or by a coefficient. Multiply by negative 2, multiply by 10, whatever. So there are other choices. Let's say what if I call the next choice 2 a q. I'll just double everything and get 2x cubed plus 4x squared minus 14x plus 8. We've actually talked about these problems before. And the only difference in these two graphs is that q is just stretched more than p is, because we doubled it. So it's stretched vertically top to bottom, but it has the same intercepts. OK, well now suppose we change this problem to one that we haven't seen before. And I'm going to give you a complex number root. So this time I'm looking for a function p of x. And let's say this one is of degree 3. And suppose that its roots are, let's say, 1 and i. 1 and i. You notice I've only given you two roots there. So you could manufacture a third root on your own. So it looks like one of the factors would be x minus 1. Another factor would be x minus i. And another factor, for the sake of argument, since I didn't give you a third root, suppose I put in, let's say, negative 1, which wasn't given, but we'll just make that one up. So this gives me x plus 1 right here. Look what happens when I multiply this out. I am going to get a cubic polynomial, but the coefficients are going to be a little unusual. If I take x minus 1 and x plus 1, that product is x squared minus 1. And I multiply by x minus i. So this gives me x cubed minus, there's a minus i x squared minus x plus i. What do you notice about the coefficients on my polynomial answer? They're not real. They're not all real. Now there's a 1 here and there's a negative 1 there. Those are real, but this is an imaginary number, negative i. And here's a plus i. So what if I added the stipulation of this problem that I'm looking for a polynomial of degree 3, and it has these two roots, and it has all real coefficients? I want to have real coefficients. How can I do that? Well, the way I will adjust it is I'm going to put in, for my third factor, something other than the negative 1 that I just chose, I'm going to put in the conjugate of i. Let me write down that word, because you've seen this before when you talked about complex numbers. But when I say the conjugate of i, when I say i, that's 0 plus i. And the conjugate of i would be 0 minus i, like we were multiplying by a moment ago. That's going to be minus i. So from my third root, I'll put in a minus i right there. And now I'll have real coefficients. Let's see how this works. I'll put in a x plus i. So when I multiply this out, I'm going to begin by multiplying these two factors, because it's these two that make the difference of two squares. I have my x minus 1. And what is the product, x minus i and x plus i? x squared minus i squared? x squared minus i squared, exactly. And we're going to reduce that next. And what can we replace this with? x squared plus 1. x squared plus 1. Now you see we've gotten rid of all of our complex numbers. So I'm multiplying x minus 1 times x squared plus 1. And the product is x cubed minus x squared plus x minus 1. Now I have a polynomial of degree 3 with real coefficients.  So if you put in the conjugate pair, you'll go back to the original real coefficients. Let's take another problem with different roots and see how we could construct that. Let's see. Suppose I'm looking for a polynomial this time of degree 4. And suppose the roots of the polynomial, or the zeros, are 2 and 0 and 1 plus i. Then I'm going to need to include an extra root with that because I said degree 4. I need an extra root. And if I also add the attachment that I want real coefficients, then what extra root should I put there? 1 minus i. 1 minus i. That's the conjugate of 1 plus i, exactly. So my factors would be x minus 2, x minus 0, x minus the quantity 1 plus i, and making a little more room here, x minus the quantity 1 minus i. Now, to reduce this, first of all, x minus 0 is just an x. So let's just put that out in front. And then here's my x minus 2 factor. I'll bring that down next. And for the other two factors, I'm going to distribute the negative and write this as x minus 1 minus i and x minus 1 plus i. Now, these first two are very easily multiplied together. That will give me x squared minus 2x. But to multiply the other two, I'm going to regroup the terms.  So you see what happened. Originally, I had the 1 and the i's grouped together. I distributed the negative.  The reason I do this is because this looks like the difference of two squares.  So when I multiply those together, I'm going to get a squared. That'll be the x minus 1 squared minus b squared. That's the i squared. In fact, I don't need that much room. Well, at this point, I'll get rid of the i's. I have x squared minus 2x times, what's the square of x minus 1? x squared minus 2x plus 1. Plus 1, right. And then minus i squared, I could write as what? Plus 1. Plus 1. So this is x squared minus 2x times x squared minus 2x plus 2. And you notice there are no imaginary or complex numbers left. I got rid of them when I squared the i squared in my last step. So now if I just finish this off by multiplying this out, I get x to the fourth. That's this product. How many x cubes will there be? Let's see. I'm going to get negative 2x cubed here and negative 2x cubed here. So that's a total of negative 4x cubed. How many x squares? Well, let's say I get 2x squared there. I get 4x squared there. Do you see any other x squares? I think that's it. So what would we say 2 plus 4 is 6x squared. And now let's count the x's. I have negative 4x there. And I think that's all of them, just negative 4x. So this is a fourth degree polynomial. It does have four roots, 2, 0, 1 plus i, and then the one that we created, 1 minus i. And it does have real coefficients. So we've been able to go back and reconstruct the polynomial with real coefficients, even when we had a complex root given. Let's go to the next example on the next graphic and look at another problem about polynomials with real coefficients. It says factor the following completely. And there are two polynomials given. Now when I say completely here, what I mean is to factor these down to linear factors. No quadratics, just linear factors. So let's take the very first problem. I'll write it here on my green screen before we take the graphic off. p of x is 3x to the fifth plus 24x cubed plus 48x. OK, let's look at this polynomial and see how we would factor it. It says factor completely. And what I mean by that is to factor it so that every factor is a linear factor, like x plus 2, x minus 1. Well, do you see any way we could reduce this, first of all? Take out an x. Take out an x. Is there anything else I could take out? A 3. A 3. Let's take out a 3x. And that gives me, well, who could tell me? What will I give you? I'll give you x to the fourth plus 6x squared. Plus 8x squared. It's 8, OK. Plus, jeez, I don't know, 48 to the ninth or 3? 12, 16, plus 16. Plus 16. OK, now, if we're lucky, we would be able to factor that without going to the rational root theorem. Can anyone think of a way to factor x to the fourth plus 8x squared plus 16? It's perfect square again. It is a perfect square. Very good. Jenny, what's it the square of? x squared plus 4. x squared plus 4, yeah, x squared plus 4. In other words, I have that factor twice. So far, I've been able to avoid using the rational root theorem and Descartes rule of signs because that's my alternative, but it would take longer. You know, if I write that factor separately, x squared plus 4 and x squared plus 4, then let's see how I could factor that to linear factors. Anyone have an idea how you could factor x squared plus 4 if you're allowed to use complex and imaginary numbers? I would set it equal to 0. OK, let's set it equal to 0 over here, and then do what? Subtract 4. OK, x squared is negative 4. And then take the square root of both sides. OK, so x is going to be, now we're looking for every possible root, so I'll put a plus or minus square to negative 4. That's exactly right. And what is that going to be? Plus or minus 2i. Plus or minus 2i. Plus or minus 2i. So I have two roots. One of the roots is 2i. The other root is negative 2i. So how does that tell me what the factors of this guy are going to be? x plus 2i and x minus 2i. OK, now you said x plus 2i. Which one of these roots were you using when you said x plus 2i, Steven? Negative 2i. Yeah, so if you take this root, it's x plus 2i. It's x minus that root. Why is that? Well, in the problems we worked before, if you remember, when we had a root of like 1, the factor was x minus 1. So this time, if the root is negative 2i, I take x minus negative 2i, which is x plus 2i. And then the other factor, Steven, would be what? x minus 2i. x minus 2i, yeah. I just want to make sure that everyone understands that when you look at 2i, that's not the one that goes with x plus 2i. It's the one that goes with this. You see, if I said what is the 0 produced by this factor, you'd set that equal to 0, and it would say x equals 2i. But now I have more roots produced by this one. But they'll be the same as this. So why don't I square this factor, square that factor? And so the factorization is 3x times x minus 2i squared times x minus 2i squared. I have five linear factors. When I say linear, I mean first power on x. Five linear factors for that polynomial. OK, we were able to get by that one using just fundamental factoring techniques. Now let's go to the other factoring problem that was on that screen, that graphic. This one I think we're going to have to go to the rational root theorem. It says q of t, q of t, I just went to the variable t just to change names a moment. And that was, let's see, can I see that once again? t to the fourth plus t squared minus 6t plus 4. OK, I don't see any obvious way to factor that. So instead, let's count positive roots and negative roots. And negative roots. Now when I say positive and negative roots, I'm talking about real number roots, not complex or imaginary. How many positive roots could we have? Two or zero. Right, sign change, sign change, two sign changes. So either two or none. Two or none. And for q of negative t, that'll be t to the fourth plus t squared, because when you square the negative, you get a positive t squared, plus 6t plus 4. Well, this time we don't have any sign changes. So there are no, there are no negative roots. So either two or none for the positive roots, no negative roots. That helps me to reduce the possibilities pretty quickly. What are my choices for p in this function? Well, p would be a divisor of 4. So that would be plus or minus 1, plus or minus 2, plus or minus 4. And we'll be restricting this to positives in just a moment. And q would be divisors of the coefficient of t to the fourth plus or minus 1. So for p over q, if I list only the positive roots, 1, 2, and 4. OK, let's take these one at a time. 1 divides into 1, 0, 1, negative 6, 4. And I'll be adding here. 1 added to 0 is 1. 1 and 1 is 2. 1 times 2 is 2.  So very quickly, we get a 0. So one of the roots is t equals 1. Now, shall I move on to 2 or should I test 1 one more time? Do it again. Let's do it again. Yeah, because it could be a root of multiplicity, too. So now I'm going to divide 1 into 1, 1, 2, negative 4. Because this is the polynomial that remains. 1, 1 times 1 is 1. 2, 1 times 2 is 2. 4, 1 times 4 is 4. Yep, sure enough, it's a root a second time. But what's even more important is not only have I found that this is a root of multiplicity 2, but I have my remaining factorization, t squared plus 2t plus 4. Now, I'll need a little room to write here. So let me just back up and take out this first portion. And here's what we have found. This factorization is t minus 1 squared times this polynomial, t squared plus 2t plus 4. Now, I don't believe this would factor by ordinary factoring techniques for real numbers. So instead, I'm going to go to the quadratic formula to factor that. t will be negative b plus or minus the square root of b squared minus 4 times 1 times 4 all over 2. So t will be negative 2 plus or minus the square root of, let's see, 4 minus 16 is negative 12 over 2. So t will be negative 2 plus or minus, how could I write this as an imaginary number? Negative square root of 12. I root 3? Let's see. Well, now you're going to have to take a square root. Yeah, there's a 2 times the square root of negative 3. So there's the square root of negative 3, and here's the square root of the negative 1. Now, sometimes people put the i over here on the right-hand side, but if you're not careful, that square root bar may look like it goes over the i. So what a lot of textbooks do is they put the i in front of the radical, so there's no doubt that it's outside the radical.  So when I go to write my complete factorization up here, I can now factor this last quadratic using these roots.  Well, that looks messy, so I think what we need to do is to distribute those negative signs. We have t minus 1 squared times t plus 1 minus i square roots of 3 and t plus 1 plus i square roots of 3. I have two linear factors here. I have a linear factor there, t plus or minus a complex number, and another linear factor there. I actually have four linear factors. That's called the complete factorization of that polynomial. Now, let's just go to the second graphic, the one on the cubic formula. Let me just mention this in passing. We only have a few minutes left here. But just as a side note, you won't have to know this for an exam. So if this is too quick of a discussion of this last item here, don't worry because it won't be on the test. But I just thought you should know that there is a formula for factoring cubic polynomials of the form p of x equals x cubed plus px plus q. You notice that polynomial has no quadratic term. It has a cubic term, a linear term, and a constant term. And to solve that, there is a formula, something like the quadratic formula. But this would be a cubic formula. And it says x is equal to the cube root of negative q over 2, q being the constant term, plus the square root of q squared over 4 plus p cubed over 27, plus an almost identical radical added onto it. What's the only change in the second radical? Second negative. There's a negative sign in front of the square root. And so there's an interesting history of this, but I don't think we have time to go into the history. OK, so this time I'd like to try solving this polynomial, which is in the form that was on that graphic. There is no quadratic term. p is negative 3 and q is 2. The formula says I should take the cube root of negative q over 2 plus the square root of q squared over 4 plus p cubed over 27. And then add on the cube root of negative q over q minus the square root of q squared over 4 plus p cubed over 27. Now when I reduce that, this is the cube root of negative 1 plus 0. And over here, this is the cube root of negative 1 minus 0, the square root of 0. So that's the cube root of negative 1 is negative 1. And the cube root of negative 1 is negative 1. We get negative 2. Now if you substitute in negative 2 for x in the original polynomial, it'll make it 0. I'm not going to ask you about this formula on a test, but if you have an opportunity where you'd like to use it on an exam, you certainly can. Well, let's see. We've talked about the fundamental theorem of algebra and complex roots today. We'll see you next time for episode 13.


Zip it What's up money geeks mr. V here welcome to another video and this one we are going to talk about zip recruiter So this is a company that Bring together employers and employees to connect them So it's not like a marketplace if you're looking for a job you can go there In post your resume and create a profile and then employers pay to use the platform So they can find the requisite skills that they're looking for in employees So the company is based out of a Santa Monica California so this company the reason why we're talking about is because it's just recently went public So the objective here is to see if this is one that we should be looking at to buy or Maybe just wait. So that's what we're gonna cover into this video But before we get started guys if you're new to the channel We took about how to earn money how to save money how to invest and build wealth So that's something that interests you go ahead and hit that subscribe button and the notification bell So you don't miss out on new content. So again Zip recruiter is kind of like an online marketplace Where job seekers and employers get, you know come together and so the company You know charge the P employers to get on the platform to use their database of Possible employees to fill positions in your company So again, the business model there is very straightforward You would see competitors like LinkedIn Doing the same thing where you connect people Together LinkedIn is even better more so because they're making it more so for social media platform For professionals where you can connect with people like the old good days where you have to go to happy hour to connect These linked in you can connect with people just like that create a profile. So that's a Competitor for them. So again, like I said, they recently went public here. So take a look here So this is our one day. Let's look at a five-day. So this stock actually opened every day one month You can see here one month. That was May 26 open up at $21 and ever since then I kind of know went up 20 21 and the since seen a high of close to 24 66 so that's almost about like 16 17% Since they went public, but again, so the question is should we buy this one or should we wait? Before we make that decision, so I jump in and just take a look at their finance their financials here So if you look at the company's their revenue for 2019 that's 429 1559 for 2020 it was 418,142 so if you look at the friends there Revenue actually dropped in 2020 we can attribute that to people not looking for work people not Employees not hiring because of COVID But other than that, I don't see why the revenue should drop so they could justify that as a result of COVID so if you look at that gross profit here gross profit was about 374 thousand three hundred seventy-five thousand four twenty nineteen and 363 364 thousand for 2020 and again that number could be as a result of COVID But I'm not super impressed with these numbers because we're talking about in the thousands Unless I mean these guys really plan to do some drastic change to really improve on that revenue slash Profit I don't think that it's a really good You know revenue for me to be excited about the company. So Yeah, is it recruiter? I think it I mean they have a place in the market. It's just gonna be a home run Is this we want to be it's gonna be that stock where you buy it and it's gonna make you a millionaire tomorrow I don't think so This is just gonna be one of those average stocks So if for right now, I'm not gonna be buying this one I'm just gonna pump the brakes on this one and see Where it goes. I don't really see because again, like I said the companies like LinkedIn have more In the marketplace than this guy and then more again people are finding work Through what a mouth and that doesn't negate what they do But personally I don't see an area where they are gonna grow and expand massively to Increase their revenue increase their profitability. I don't really see that again That's my personal take so you can do your own research and decide what you think about it So I'm not gonna be buying this one. I'm gonna put this one on hold To see if they come out with some more information now that they're trading publicly Because they're gonna have a ton of pressure to increase their revenue or increase their profit from their shareholders. So I'm gonna see how they, you know Show that they have a roadmap to really make some changes to increase revenue and profit profitability before I'll jump in on this one. So that's all I wanted to cover in this video again LinkedIn is now trading publicly under the ticket symbol here ZIP that's the ticket symbol if you want to look it up the ZIP and check them out So, I mean they're up now about almost 17% from their opening IPO day But again, you would see some sort of pull back here pretty soon because I don't Really think this one has room to run Massively so again, let me know in the comment section what you think are you gonna be buying ZIP recruiter or you're just gonna pump the brakes as well As I'm doing here. Let me know in the conversation if you're new to the channel We check about how to earn money how to save money how to invest and build wealth There's something that interests you go ahead and hit that subscribe button and the notification bell to you don't miss out On your content also guys if you are looking to get started investing We both has their promotion where if you sign up and deposit a hundred dollars You get some free stocks links in the description and as always guys do your due diligence Don't be a greedy savage and stay motivated


Hermippus, my friend. I once heard someone observe that ancient philosophers conducted most of their teaching in the form of dialogue, but that this method of composition has seldom been practiced in more recent times and has rarely succeeded in the hands of those who have attempted it. Accurate and logical argument, which is now expected of philosophers, naturally leads to a methodical and didactic style that immediately explains the point without introduction and then proceeds without interruption to deduce the proofs on which it is established. Presenting a philosophical theory in conversation is seldom natural. The dialogue writer, by departing from the direct style of composition, desires to free the performance and avoid the convention of author and reader, but encounters instead a worse difficulty, that of conveying the image of pedagogue and pupil. If the dispute is conducted in the spirit of good company by introducing a variety of topics and preserving a proper balance among the speakers, so much time is lost in preparation and transitions that the reader will hardly find it is worth sacrificing order, brevity and precision for the graces of dialogue. However, there are some subjects to which the dialogue form is especially well suited and where it is preferable to the direct and simple method of composition, any doctrine which is so obvious that it scarcely admits of dispute but is so important that it cannot be too often repeated seems to require some such method. The novelty of the style may compensate for the triteness of the subject, the lively conversation may enforce the precept and the variety of lights shed by the various people and characters will appear neither tedious nor redundant. Any philosophical question that is so obscure and uncertain that human reason can reach no agreement about it, if it is treated at all, seems to lead us naturally to the style of dialogue and conversation. Reasonable people may be allowed to differ where no one can reasonably be certain. Opposing opinions, even without a decision, provide delightful amusement. And if the subject is unusual and interesting, a book carries us in a way into good company and unites the two greatest and purest pleasures of human life, learning and friendship. Happily, these circumstances are all to be found in the subject of natural religion. What truth is so obvious, so certain as the existence of God, which the most ignorant ages have acknowledged, and for which the most refined geniuses have ambitiously endeavored to produce new proofs and arguments? What truth is so important as this one? The ground of all our hopes, the surest foundation of morality, the firmest support of society and the only principle that ought never to be absent for a moment from our thoughts and meditations. But when we treat this obvious and important truth, what difficult questions arise concerning the nature of that divine being, its attributes, its decrees, its plan of providence? These have always been the subject of dispute among people. Human reason has not reached certainty about them. But these topics are so interesting that we cannot restrain our restless inquiry into them, though nothing but doubt, uncertainty and contradiction have so far been the result of our most careful research. I had an opportunity to observe this, when, as usual, I spent part of the summer with Cleonthe's, and was present at the conversation he had with Philo and...