 So if fx is divided by ax plus b, the remainder is f of minus b by a, right? And if ax plus b is a factor of fx, f of minus b by a has to be 0, correct? We have also said that number of, we have learned the concept of degree, so you know what a degree of a point on the list, so questions will be related to that. So please be careful about the polynomials themselves, whether they are polynomial or not. Second is, what else? Yeah, number of roots is equal to degree of a number of zeros is equal to degree of polynomial. Then a zero of a polynomial is that value of the variable which reduces the polynomial to zero, correct? So hence, be careful while you are dealing with the questions because the words could be a little misleading. Okay, so can we start? So all of you there, anyone wants any other assistance? So let's start, I hope. Again, one point I missed was, all of you have to keep track of your accuracy, okay? So it would be a good, I would prefer that I would recommend that if you maintain a notebook of centymar questions and your, you know, learnings as well and accuracy is very, very important. Okay, so let's start. So first question on the board, guys. Oh, there is some change here also, shortcut enter, start quiz. Very, very simple question. I hope you get a good warming up. Sir, the question is not coming from me. Why so easy? If I give you a hard question, you say, sir, why so hard? If I give you an easy question. Sir, the question is not coming from me, just coming. First question has to be, so bonus, everyone. Sir? Yes. Hello? Yes, sir. Tell me. Tell me, who's speaking? Yeah. Sir, I can't see the question. You can't see the question on your screen. It's available on my screen also. I am sharing my screen. You can check it here. Why so many pizzas? Okay, you want pizzas? That's all. So this is Pratik, sir. I can't see the question. Oh, so look at the question on my screen. Who's speaking, by the way? Look at the, if you are not able to see the question on your, this thing, look at my screen. I have already shared my screen, right? It's available on my screen. Yes, and we correct. I have to start a pizza shop myself now. Okay, so the first one, I hope you are cool. So most of you have got it, right? So do I need to explain? Haritha, quickly change the system or some other device, maybe. Okay, so anyways, what was the explanation? Yes, so never mind. I have a full-fledged PowerPoint here. Yes. So this was the question. The zero of the polynomial fx is 2x-3. We know to do what? 2x-3 has to be equated to zero. This color I don't like, so I will change it. Wait a minute. So color pointer options. Then highlighting color white. Okay, that's fine. So hence 2x-3 is equal to zero. So x has to be 3 by 2. And that's the answer. And here is 3 by 2. Cool, right? Let's go to the leader. Okay, here is the first leader board. I know you know who ya boi has the highest score. Then Tanvi, Vihan, Manisha, Tanya, Karun, Flowerboy, Tony, Flowerboy. Oh my God, you started setting flowers. Pratham, Vihan, and Nishant. Okay, cool. So this is... Sir. Yes, sir. Sir, it shows I'm in 10th place but I don't know. I have only 979 points. Oh, is it? Never mind. So you are just below 985 Bharat. So next question. Okay, so hence top 10 people. Next question. So it is so close. First question is so close. Okay. Where is the next question? Where did it go, by the way? Next question. Okay, so yes. Question is easy. Be careful. Don't repeat the mistakes you have been doing so far. So don't miss click it. Don't be early. Look at the question. Try to multiply. Sir. Sir. Sir, it says that I'm 5th. Like last question it says that I'm 5th but I'm 8th. You are 5th. You are 5th Tony. Don't worry. Okay. 100% Tony. 4th place. That's why. No problem. All of you maintain your accuracy. Chalovi. The names Tony Tony. Okay. 55 more seconds. I hope you could sense the issue. Okay. Okay. Trap master. Who's that? Yes, I sense the trap though. You sense the trap. So you have. Sir, I saw the trap. But then I was about to miss click. And then I finally in the end moment saw it. Can someone miss click by the way? It's so difficult to miss click intentionally when you miss click. You have done it by. I was about to click some other answer. That's better. Yes. Okay. Cool. 10 seconds. 8, 7, 6, 5, 4, 3, 2, 1 launch. Plus 1. Plus 2. Plus 3. That's nice. Okay. Sir, I got it. If somebody thought there was a trap and thought they were observing to see it. So here was the question guys. So question was very simple. You had to simply see that this into this will give you actually I had made some more you know the question was a little difficult but then anyways so I changed it and if you open this you will get a term with X to the power 4. So that's the higher power highest power term you will not get anything else and this fall short by 1. So hence the degree is 4. So C is the answer. Right. Anyone didn't understand this? Sir, I little wasted so much of time looking for traps. Sorry. I could not get you R. What did you say? Sir, I wasted so much of time looking for traps when there were no traps. That itself is a trap. Nice. So now you have to be you know Sir, you have carried so many traps that even if no trap people will look. So that's I think I over sensed the trap and I marked it wrong. You need to be balanced guys. Come on. You can't. Second question anyways. Good that you are conscious but at least you got get it right. Did you get it right? Then it's fine. No problem whether there was trap or not trap. You should get it right. Okay. Cool. So we have two questions on first position. Then we have Tanya, Pratam, Bihar, Nishant. Tom. Who's Tom? Tom my dear. Who's Tom? Cool name. Tom. Ashish is Tom. Okay. Top hat. Ashish. But there are two Tom's. Sir, I'm not Tom. I'm not Tom. Sir, I'm top hat. So who's Tom? Tom is who? Sir, it's Roshini. Which Tom? No, Roshini can't be Tom. Okay. Next question. Roshini is light. Roshini is not Tom. Cool. Here is the next question. Now you must understand how to find out the value of a polynomial. So hence P this calculation can be a little misleading. Be careful with the signs. So the signs appear to be a little confusing. Deliberately so. So please check here. Monisha is cursing me. I take her curses there. But the day when you will succeed in any exam, then you will call me up. Correct? Don't worry. Curse me as much as you want. But then you will win. Don't worry. Be careful with the signs. Take time. Yes. Easy questions are more or you know harder. Yes. Tanvi says O n triple O. Why are you giving so easy questions today? Something is wrong with something sir. Something I can sense daily. Why are you guys Why don't you think like this that you have improved? Why do you always want to be in this lower side? Because easy questions then he will put I get it. You have already improved folks. Why do you keep yourself in a lower light? All of you are improving. Hence you are finding now easy. Good. Cool. This is the result. People are improved in mathematics. That's cool. Very good. Sir I took time. But don't make a mistake. Okay. Very easy question actually. So f of minus one. What was the figure all of you got? F minus one. How much? Very very simple. Minus one. I'll tell you this question is related to something called odd and even polynomial. You have heard of odd numbers. Even numbers there are polynomials which are odd and even as well. So this is a case of that. Anyways let us first find out. How much? 20. What about f of one? You got 6. 6 and then 12. So f of 0 is 12. Add 12 to 6 18 and minus 6. 20 minus 6 is 14. So this is this. Now I was talking about odd and even polynomials. You will study later. So if f of x is equal to f of minus x then these type of polynomials are called even. And if f of x is equal to minus f of minus x then these type of polynomials are called odd. So wherever there is a power of 2 it's always an even. So let's say if I say fx is equal to x square and f of minus x also will be x square. See fx and f minus x are same. So this is even. Whenever you subtract these kind of if you subtract even functions you will get 0. If obviously they are the same then. So hence if you have seen x square term will just get cancelled. You don't even need to calculate x square right. So that was the some information to all of you. Anyway so this was a very easy question. Let me see who did what. So ok. Prasam strawberry naffle 1 is no it's not 1. So Vihansingh NPSHSR cool. Very good guys. Srijan was, Manisha was fastest. Manisha you are in third position. Good. Let's go to next question guys. Question on board. Take time. You guys have improved. So don't worry about I just gave you the hint in the previous question. I never knew that this was the next question. Ok. So please solve this one. So be careful with the powers which is given. Be very very careful with the powers. The series choice of numbers have some some influence on the answer. And I have a very strong belief all of you can find the right answer. Shivam has found the trap. Good. 5, 4, 3, 2, 1 and satellite rocket has been launched. Very good. Ok. Ok. Ok. Ok. Let me see there. Now the results are coming down. Ok. Very good. Anyone who didn't understand this question this was very very easy question. It was very easy question. So fx is simply this simply this right? f of minus x will see what will f of minus x be f of minus x be minus x to the power 9 minus x to the power 7 minus x to the power 5 minus x to the power 3 but x squared. So this was this correct and you have to add these two. What was fx? fx was simply x to the power 9 plus x to the power 7 oops 7 plus x to the power 5 plus x to the power 3 plus x squared. So you know you guys are wise enough to sum it to D. Ok. Clear or not guys? All of you are clear? Why would the x squared remain as it is when it is f minus x? Because minus x whole square is equal to x square only. Yes I got it. Yeah. So hence what is the learning? Negative square is always same right? Whether it is x square or whether it is minus x whole square same, x square even polynomial even polynomial. Actually it is an even function but cool. So hence if I would have given you 1 instead of x to the power 2 it was x to the power 1 then the answer would have been what was the answer if x to the power 1 was there? Zero. So you would have got this cool. Next. Ok. So the most of everyone from the top 10 is getting the right answer. That's good. Monisha is now on number 1 position. That's cool. Monisha. So let's go to Goti-Mangrahesh Karun was fastest. Ok. Next question. Ready, steady go. Zero of a polynomial is that value of the variable which reduces the value of the polynomial to zero so easy. So last 5 4, 3, 2, 1, go. Let me see the result. Ok. Ok. It is gradually going down gradually going down gradually going down gradually going down gradually going down gradually going down. Ok. So most of you have done the same mistake. I think you have not seen the 2 little sweet 2 here. So all of you have just simply put 5 into as the right. So this was not the these were not the roots. So you have to first expand it and you will get x square minus 7x plus 10 when you expand it there was a small 2 over here. So you have to add this as well. And you will get x square minus 7x plus 12. Ok. So this is your original fx which when you split the middle term you will get x square minus 3x minus 4x plus 12 which you will get as x minus 3 and x minus 4. Ok. So here is so what are the answers those who wanted to solve it. What did you see the moment I press the next button I told I gave you the definition of 0. You could have simply put the values. Clearly if you put 2 here it's not becoming 0 right. It's not going to be 0. fx will be 2 right. So clearly this is ruled out that's what you do it but then this is ruled out. Similarly this is ruled out. This is also ruled out only possibility would be the you can just check one one of them ok. So sorry guys you need to be a little bit more careful ok so let's go to lead a board this is something new ok. why my god so hence careful guys you all know it so it wasn't I didn't forget to change the signs so I just made a sign mistake while always remember whenever you find the roots the signs will change ok always remember so this learning should be there forever that whenever you are trying to find out roots the signs whatever you see in the expression will change so that's a learning so that's a check point also ok next question I made one question and wrong and may have went from 13 to 25 see what it is so ruthless Arushi I told you my friend one of my colleagues we lost J.E. by one marks one one single one single digit one one year wasted now I hope you have studied something called Pythagoras uncle's theorem Pythagoras was a person who existed in 6th century BC so this man is troubling us since last 2500 years plus plus yeah more than that so you guys are going to get your degrees because of this guy who has done something in what 6th century BC when Gotham Buddha was roaming around our planet he was giving life teachings to all these people ok ok very good so let's see how many of you have got it correct now ok ok ok ok ok cool so here is the result which is like first time the correct answer has gone down a number of years why I don't know so it was that hard it was very very simple is it so where do I draw it ok so let me draw it here so this is the triangle let's say this is h minus one and let's say this is x we have to find out the x so h square is equal to h minus h square plus x square this is the you know the equation or relation which Babylonians have been using pyramids have been made out of this so you can't make mistakes on this so x square is h square minus h minus one whole square and here is where our a minus b a plus b the sweet little identity is going to be used so a minus b first and then a plus b isn't it so this guy becomes one this item is one so this is reduced to 2h minus and you have to find out square of the third side so you have to find out h square only sorry x square so x square is 2h minus one so how many people got this right can you show that slide 18 people nice nice why is that nice I don't know anyways I'll enter leaderboard in this question now oh because of that leaderboard is something which people are looking for ok so raga hsr yid is there in the leaderboard what is yid I've heard of yid what is yid what is yid oh is it so or when you're just throwing something oh yeah so for example if you were to deep in a state someone would say why are you just eating me out of the window everyone is wrong yid is a state of mind it's not whoever said that is i got it whatever is the yid I don't eat I eat ok next ta ta common sense question the underlying concept is common sense I didn't even click on an answer and it's showing the your answer page ah who's this Karun what happened what happened so if I didn't even answer the question and it just took any I don't know what it took it came to that your answer page oh Pradeek are you on a mobile phone no sir I'm on the pc oh that happens at times Pradeek but how what's your current strike rate sir strike rate how much now I think this will be wrong so it will be like 5 out of 7 ok yid 7 out of 7 sreejan dada anyone else 7 out of 7 karun menon darum very good raga pande 6 by 6 where is 7th question 6 by show 7 roshni nice cool sorry oh ok like that cool waiting for the answer to be correct first everyone has voted and this is the answer oh my god good nice you didn't get it why did you not get this question this was very easy people were saying easy is it 0 one root is 0 one root will be 0 no excuse me sir sir I think what went wrong some people they did see the x of them but then they just thought that the product of the 0s will be 2 because only 2 0s will be their x if the other 2 has 0s then x won't need to be a 0 they must have gotten confused sir because I almost got confused you didn't understand the confusion come again once again you didn't understand why would people be confused because this is our correct answer I was probably wrong this is qubit polynomial so there are 3 0s is it this is these are 3 0s there are 3 0s to it why 3 0s because the degree is 3 qubit polynomial it's not our quadratic polynomial okay okay so hence it will have 3 0s now product of 0s so all 0s it's quite evident from the polynomial itself that the 3 0s this particular polynomial can be written like this x minus 0 and x minus 1 and x minus 2 so what all are 0s so anything which reduces this polynomial to 0 is a 0 of this so what all will reduce it to 0 x is equal to 0 will work x is equal to 1 will work and x is equal to 2 will work so product if you multiply them you'll get b right understood anyone didn't understand we'll explain again guys once again what is the degree of this polynomial 3 can you see that how many people did not see this x that's why this x was given that you fall for it I have deliberately put an x over there so even if you want to curse me I don't mind because that's how it is going to help you so you should be careful that this is going to be x into x minus 1 into x minus 2 so so how to find out 0s I told you in the class how to find out 0s if you want to find out 0 of a polynomial you simply equate it to 0 okay now when can this entire polynomial be 0 there are 3 possibilities 1 where x is 0 itself the moment x is 0 don't you think the entire polynomial will reduce to 0 so x is equal to 0 is 1 0 similarly if x equals to 1 if you put then also this will reduce to 0 so that is also a 0 then third is x is equal to 2 if you put this also this polynomial is going to reduce to 0 that means there are 3 0s so how you should have approached this sir sir I realized that I answered like I realized what the correct answer was after answering the question so should I include it in my accuracy or not no once you have marked it the die has been cast my dear so that's what you know so I also realized that I have made so many mistakes in my life and I know I shouldn't have done that but that will not change me I should count that as mistakes only okay is that clear now the person who had asked me this question any doubt y y y y y y brush don't spam y y y y or n n n which is x x whatever come on come on come on come on come on come on quick all of you let me see how many of you there everyone is just a passive viewer all of you should write y okay very good very good nice next question let's go to read the board is here a lot many people did not do it correctly very good and the top first is at his position okay next question please this question is based on remainder theorem hint remainder theorem suggests that if ax plus b or if fx is divided by ax then the remainder is f of minus b upon a a wish I buy please don't do that I am looking at the chat box so hence please do not write only something which we can relate to or which we can understand 17 questions are there totally 5 4 3 2 1 this question you have done in numbers lot many times I think you should be able to solve this one okay good so majority of them have got it correct but then most of them have just marked it by guesswork okay so how to solve such questions so what should be added 2 fx equals to this so that the sum is divisible by x plus 1 so you know that if you remove the remainder or whatever is the remainder if you just remove that remainder you will get the solution is it so what is the remainder for this simply f of minus 1 is the remainder f of minus 1 is the remainder is it remainder is so if you divide if you really divide fx by x plus 1 the remainder you will get is f of minus 1 which is 2 times minus 1 to the power 5 minus 4 times minus 1 to the power 4 minus 3 times minus 1 to the power 3 plus 2 times minus 1 minus 5 this is the remainder so minus 1 to the power 5 odd power will give you negative 1 so hence this will become minus 2 then minus 4 then plus 3 then minus 2 minus 5 minus 10 right yeah so minus 10 is the remainder now yeah so now what should be added to this so that this expression becomes divisible by x plus 1 so this is the remainder you have to kill this remainder remove this remainder what should you add now add 10 to it the remainder will now become 0 and hence the expression will be divisible by x plus 1 clear is that understood by all any x any anything any problem sorry sorry variable and equating it to 0 did you understand come on how did you solve it did you could you get the answer you equated this to this to 0 the whole equation minus 1 minus 4 and all this I added 1 k to it and then equated it to 0 no then you are reducing fx to 0 but the question was not to reduce fx to 0 or you added a kr that okay so you are saying in this you just simply add k like that yeah no that way also you will get k is equal to 10 correct no right right understood that's fine that's fine yes sir I did another method I mean it was a long but basically I took option and saw if the remainder 0 no Suneena that will be very very troubling thing because there is a power 5 so you have to judge whether this is doable things way to the power 5 you multiply and then power 4 and 3 and all that no in this case going from options it's not justified okay so hence the understanding was remainder theorem what is remainder theorem fx is divided by is divided divided by what? ax plus v then remainder is how much? f of minus v upon a remember this always next let's see the leader board hmm okay so some shuffling is happening so that's nice okay so this is question number 8 done next question guys calm stay calm we are in 2020 so 2020 is getting our lives in all our lives 2020 is in our lives we are going to remember this forever is it till we are on this planet you all will be giving better it as a COVID 19 year so you will be giving so lots of GAN to your next generation on this year right done as much as you can giving this year yeah yeah there are lots of lots of viruses now so there is a news that says China is now even making t-shirts and caps saying banned China products yes sir so it means there are many more viruses and they are selling those things to Indians that's because the large majority of China people work in factories sir and they don't even know what they are making and why they are making so they are just making it sir tick tock the banned oh oh Tony is affected Tony you had a chat it should be sir in the question so you are giving p of x and then in the option is f of x so now that I open that anyway you will understand the sense right so how many could get it 36 cool that's nice so you would have figure it out that x plus 1 so if if you put minus 1 into sorry this one is fx so mistake so fx is this so is divisible by x plus 1 that means if you put minus 1 to the power 20 20 and all that it should give you 0 correct this is understanding and which you can see it's clearly giving 0 because all are even powers even powers and they are very much paired so this will get cancelled similarly this will get cancelled and this will get cancelled all of them are getting cancelled now it doesn't matter whether it is plus 1 or minus 1 right in both cases since all the powers are even all the powers are I'm sorry so the answer see an option see yes both of them okay okay anyone wants more clarification on this so this is factor theorem application of factor anyone wants an explanation on this guys quick tell me no okay can we move forward move ahead go away no one is saying anything okay let's go do sir can you just show the screen once more where I wrote it here I wrote it so all are even powers whether you put 1 or minus 1 to find out let's say f of minus 1 or f of 1 both will be going to be both are going to be 0 correct so hence both are true x plus 1 is also a factor thus language was a little different that's it next so prasam strawberry with lots of capital letters in it is on top okay next sir the system randomly took an answer before I could even click you know the right answer no yes sir no problem questions are more important than the leaderboard sir your screen is stuck at least for me you have to refresh it Tony my dear friend okay okay so here is okay oh my god oh my god oh my god why what happened people could not get it correct okay so this was the question fx and gx are two polynomials having x square minus minus common factor if x x square minus 9 is a common factor then x minus 3 and x plus 3 are the factors right all well all good right so x minus 3 are factors correct if x minus 3 is a factor then you know that f of 3 is equal to 0 and f of minus 3 is also 0 and similarly g of 3 is equal to 0 and g of minus 3 is also 0 all these are 0 so if you see this value is 0 f of minus 3 plus g of 3 is 0 so this is also 0 this is also 0 both are 0 below any difficulty only problem was I had not given you a linear factor then you would have cracked it very easily if I would have given you simply x minus 3 instead of x square minus 9 but since I had given you x square minus 9 so lots of people got confused and they did not complete the sum so x plus 3 was to be so there are two factors I am talking about the moment there are these kind of expressions we are talking about more than one factor okay so answer is c both 1 and 2 right is the right answer below yes but you want to you want our explanation again I am saying so can I just erase how do I how do I pointer options eraser oh I have good so I have eraser also in this so let me erase guys once again quick quick quick don't worry don't worry quick explanation so this is pointer options 10 now they go x square minus 9 is of factor okay so this is x square minus 9 is nothing but x minus 3 and x plus 3 that means x minus 3 and x plus 3 both are factors both are factors of fx and gx right that means f of 3 is equal to f of minus 3 is equal to 0 and g of 3 is equal to g of minus 3 is equal to 0 correct hence what will happen all of them are all of them are 0 this is 0 0 minus 0 is 0 this is also 0 this is also 0 0 Tanya is typing something very fast okay so understood hence all clear okay next Tanya was typing very very fast 2000 words per minute times oh very fast no you are not typing I am not typing sir no problem let's go to next question fellas come on as don't get trapped it was Monitel who was typing my god so fast typing Abhishek is listening to music Menti randomly takes answers it cannot handle more than 50 people it seems don't worry we are developing our own chatbot so you will get to see that Menti some European fellows have done this soon you will be able to play that in our app directly Monitel sir some Johnny guy right I tried writing to him but he never responds he keeps sending mail by the way lots of mails and things but then never responds okay so 18-17 14-18 12 so the reason the email comes is like when you ask for the answers and all it says by Johnny and it will give the email yes sir it has if he is only talking to us but it's not yes only 15 see people were saying easy easy where was it easy what happened in this this was the fair my god I have to again do some fasting you don't know this you can common factor factorize this factorize this this is simply x-1 x square plus x plus 1 and here itself the game is over because clearly x-1 is not a factor of this why because if x-1 is factor of gx then g1 must be 0 which is not g of 1 is not 0 that means x-1 is not a factor so what else will be game over here itself did you understand guys it was very very easy question I don't know why you made a mistake or you could not do it yep you have to simply hence this was given see many a times you try to find out some lots of gold in that mine it's not like that it's very very straight forward you should have just factorize it factorize this and you would have eliminated what that this guy cannot be a factor of gx why because if that was so then g of 1 must be 0 which is not clearly so hence this cannot be factor so what is left this is hence got it anyways so next this is interesting interesting next sir I didn't do that I did like I took the x and I then made it 2x and then minus x and then I did a square minus b square and all that why this has to be very simple don't over complicate go by exactly simple have you found out any question which requires a lot of complicated stuff every question is solvable within 10 to 15 seconds if your if your approach is correct so anyways learning last question of mcq type this would be interesting I believe let me see how many of you this is flat okay okay now mine I will solve it for you so the easier route was option route in this case easier was option try option and you get it try option and you get it okay so but anyways I will tell you the formal methodology so see divisible by x minus k by 2 that means 3 times replace x by k by 2 then minus k times k by 2 whole square plus 4 times k by 2 plus 16 must be 0 right this is what is called factor theorem so if x minus 8 is a factor then f of a must be 0 so that's what I have done this must be 0 so if you solve this this will be 3 times k cube by 8 minus k cube by 4 plus 2k plus 16 I would have come to this level and then I would have taken the options route they go so this would be 3k cube by 8 minus 2k cube by 8 plus 2k plus 16 that means it is nothing but k cube by 8 plus 2k plus 16 if you see clearly this cannot be k value because if k is 0 then this is not coming out to be 0 so this is not eliminated similarly so what was the answer d similarly this is this is also not possible all positive values of k will not hold so it was very very easy so all positive values are eliminated why because if you put any positive value in here it will never be 0 sir it shows I didn't vote in time but I voted long ago so should I come here in my accuracy you are the best judge I don't know what happens to me you are the best judge of your performance there is no point sharing something if you don't owe to someone else anyways it's neither helping you nor us so I don't mind you count whichever way you want yes there are some places I understand but you know whether it was true or false or very wrong answer could you get this guys so hence d minus 4 is the right answer but if let's say it was a subjective question you could have factored this you factorize this it was factorizable yes it required some bit of some bit of this is the this is the equation you are getting so you have to simply factorize this part and the catch for this now you have not learned this this is a cubic part so here you have to do some trick what's happening so anyways integral root theorem you can use that yes very good now that you are speaking the integral root theorem I have discussed that so you can use that or it has to be 2 right one of the things has to be minus 2 or 2 because 128 is like just 2 into 2 into 2 into 2 into 2 but only catch was since none of the k cannot be to make it 0 k cannot be positive you have to think like that so k cannot be positive k has to have a negative value so that ways this is done anyways I am just explaining how you could have done it otherwise so you have to do this 4 k separate take it then square and then oh sorry here I am writing ok so you have to break it like this so this is a different type of factorization so I am adding 4 k square and I am taking 4 k square simultaneously and then this is the this is the factorization first step now you can factorize here so these 3 you can factorize by splitting the middle term and this here you can see there is a k plus 4 common k plus 4 is a common factor so hence the common factor will come out to be k plus 4 right and when you are equating it to 0 you will get k as minus 4 I am just leaving it here you can do the factorization from here on ok so fair enough let me see the last MCQ all of you are ready with the rapid fire ok Monisha got the right answer very good Anuya also got the right answer there are 4 people who 4 Anuya and anyways ok ok ok ok next is only 8 seconds and very very easy this was not a polynomial so answer is false so the question this was not a polynomial guys you should have done this is it a polynomial it's not a polynomial right y is in the denominator so it's not a polynomial so no degree false great only 22 people got it correct that's watch the my screen don't watch your devices screen ok so some shuffling very good sir yes I might need some scam I said false and it's saying wrong answer oh my god sir that is happening to me also sir on your screen it shows me that I am on third place but on my screen it says I am 33rd who is this Grahish your score matters no it says I am 33rd place no problem no problem solve the question your entire focus is on whether I am at first position 7th position 18th position do I care I don't care I don't know so hence you should also not care because right now focus on whether I am my understanding is clear or not there are some tools which are glitchy I understand let's not care about that see again ok good that 37 people got it what was the question so question was so many people try kahoot next day kahoot is ok again my dear students till your focus is on the leaderboard only you will not learn any of these questions this all gimmickery is done only to make it little fun last year we never had quiz format we only solved problems on ppt's so please understand ok so if you are learning till you are learning the purpose is solved whether you are coming first or fourth or tenth or hundred doesn't matter to anyone right now if you are learning while you are going for the test real test then you will perform there that's best that's the best part of it so hence don't worry about this you know your accuracy levels so question was if the constant term of a polynomial is 0 then polynomial has 0 as one of the zeros we already saw that in one of the questions where x was hidden very small so this is a true statement right constant term of a polynomial will be 0 if do you understand this part yes or no this is very interesting you must know this if fx is x to the power something let's say x to the power n plus x to the power n minus 1 plus blah blah blah plus x to the power 1 plus let's say constant k ok so the constant term will be 0 when a polynomial has 0 as one of the zeros is it why because if 0 is one of the zeros then f0 is also 0 so the moment you put f0 is 0 every term will become 0 only k will be left which will again be equated to 0 so k is equal to 0 comes from here so always remember if the if the constant is 0 in the polynomial it will definitely have 0 as one of the zeros so I hope and most of you have got it correct so good next ok so Srijan and Krishna got it not great ok next this is done multiple number times you know this answer false very good right what was the question question was never terminating is a wrong wrong statement it is terminating so hence false statement sir it says you didn't vote in time the answer was true right very easy why because you should have done a minus 1 is root 3 you will get 3 correct so answer is true is true next last question of the evening guys last question of the evening ok ready steady go ok so final one this I think most of you would have hmm oh you got correct ok how many of you know genuinely solved this question I didn't vote in time so hence the server there is 8 seconds come on anyways did you my again focus is somewhere else did you understand this question this question could take you to you know very high places I don't know if you very noticed it if f1 is equal to 1 then x minus 1 divides fx minus 1 true or false true why why is this true because let's say you are defining this another polynomial gx as fx minus 1 ok so g of 1 is equal to f of 1 minus 1 which is if you see from here f of 1 minus 1 is 0 right so hence g of 1 is 0 g of 1 is 0 that means x minus 1 divides gx and what was gx x minus 1 divides fx minus 1 yep ok clear so let's see the final leader vote I will show Bharat just stay back those who want to understand any other question stay back ok ok here is the final leader board so Karun leader back very good Karun congrats Pratham Shivam Vihan Srijan and who is this R and R guy can't see him and then Monisha and Goti and Krishna and Kostu ok so irrespective of these names accuracy guys accuracy accuracy accuracy all of you please put your numbers here I want to check that I will take only your numbers Srijan 14 Srijan 14 come on all of you Karun 16 very good 14 17 13 13 17 14 17 16 17 good Pooja very good 13 by 17 15 by 17 I am sure there is a big lag in me typing so by the time I type in the class you get over so I got 16 out of 17 very good who is this Kandhar I think Kandhar it was you no sir this is Greshin you are smoothly from class ok your voices are so resembling ok so very good nice ok Arthur take care buddy so 15 by 17 is good good so those who are around 14 plus reasonably they are good commendable very good nice very good no there can be one second also so 12 17 14 17 good nice so I hope you learned something today and from next time onwards bye take care we will be now bye take care thank you so much bye good night thank you so much if you want to get any explanation can stay back I will explain the question yes yes Barat I am staying back tell me Barat which question you want to go through once again tell me Barat you can unmute and say buddy ok so the 16th one and the 17th one 17th one F1 is equal to 1 ok so can I say F1 minus 1 is 0 yes sir right let's say I am defining a polynomial gx as FX minus 1 whatever is FX FX minus 1 is gx let's say ok so if I put x equals to 1 so g1 will be how much F1 minus 1 which is 0 correct yes sir that means if g1 is 0 Barat then what does it mean it means that x minus 1 will divide gx yes or no by factor theorem if g is just the name of the function or polynomial so g1 is 0 that means x minus 1 should divide g1 yes or no gx sir I am not getting this part see if F of 2 is 0 don't we know by factor theorem you have studied factor 2 right yes sir so by factor theorem can I say that x minus 2 will divide x minus 2 divides FX yes sir similarly if g1 is 0 can I say x minus 1 will divide what gx correct so x minus 1 divides gx and what was gx my dear friend check that means x minus 1 divides this symbol is for divides divides gx which means x minus 1 divides FX minus 1 yes yes who is this sir this is true why I just received the sent them books but I have got 2 of each subject 2 identical copies of the same subject so you got what all subjects so I got all 4 subjects but 2 of each and they both are identical the contents everything same so you got 8 books is it yes sir I got 8 books totally and both of them were addressed to you only yes sir all the 8 ones I delivered to my house all the 8 books no problem keep that copy with you we will later on take it from you whenever we meet ok my dad I think he didn't see the message or something he still didn't put in the information that you wanted sir but will I still get the book if I put it now who is this Arushi yes sir why not yeah yeah ok yeah sure yes sir yeah I am going to that a is root 3 plus 1 so a minus 1 is root 3 right yes sir yes what will you get ok yeah yes sir got it got it take care buddy ok see you good night bye Janth is there too guys anyone else wants any other explanation bye bye folks take care good night