 So we are going to talk about polynomials today, right? And let's begin. So the first thing as we have been doing, so what has been removed from polynomials? So this has been removed from polynomials. So there is a statement and simple problems on division algorithms for polynomials with real coefficient has been removed. That is important in terms of the knowledge of function and algebraic function specifically. So hence, I will definitely share that statement and at least the theory part. We'll not do problems, but you must be aware what division algorithm for polynomials is, right? So this has been removed. So there will not be any question of finding coefficient if a particular polynomial fx is divisible by some other polynomial gx. And this is the equation qx and this is the remainder rx and hence find out the polynomial, that kind of questions have been taken away, okay? So that will not be there in the board, okay? But very, very important as far as the concept of polynomials and functions is concerned. Okay guys, so again, review of the structure. So this should be very, very clear again. So hence, in every slide, every PPT, I'm making sure that this is there in your mind every time. Okay, so what is it? So we will not repeat everything now. So you know, there are two parts A and B and 36 and 44, these are the two part marks. So 36 marks is only one marker, so to say. And in polynomial chapter, there is one case study question in the sample paper. So polynomial chapter will be important, right? So maybe that in the sample, in the actual paper, there might not be a case study on polynomials, but in the name of case study, they have just given some diagrams, otherwise the basic knowledge of polynomials will be good enough to solve the question. So you'll see later when we discuss that. So hence, don't worry about that, okay? So case study questions, in my point of view, what they have done is, they have just tried to make a story around it and the questions are all, whether it is linked to that or not irrespective that you will be able to solve it. So you must be aware of the fundamental, that's it. Now, as we have been discussing, so there are 36, so 16 into one, so you know that break up now. Vibhav is joining and then leaving, joining and leaving. What is happening Vibhav? Are you there? So lot many times I have to admit him, okay? Okay, now, so let's begin. So what I'm saying is 16 into one, you now remember? Then four case studies, each with five questions, but you'll be given four marks only. So this is 16 plus 16, 32, okay? And then we have six into two, mark 12, seven into three mark, 21, and five into three, five marks into three, so we write as three into five marks. This is 15. So this is the break up total is 44, sorry, 48, right? Yeah, 48 and 32, this is total 80. Okay, so let's go to the definition of polynomial. So this is the polynomial definition. What's polynomial definition? So let X be a variable. Now this is polynomial in only one variable. Polynomial could be in multiple variables. So now what we are, I'm going to do at least in mathematics. I'll give you some brief idea of whatever, what is going to be later as well. So hence now we are talking only about one variable. But there will be polynomials with multiple variables as well. So here, the typical way of representing polynomial is this, and here is where these kinds of statements are not very, you know, so people are uncomfortable seeing these kinds of statements, but you have to now get accustomed to such things. Now polynomial in X can be represented by F and within brackets X. Now this F is for function later on you are going to see. So the polynomial is nothing but there's a word function hidden all the time, which because of your, let's say lesser maturity in mathematical knowledge. So we have been avoiding this word, but it's actually a polynomial function in X. So the variable is X and you can see these are the real numbers. So we are now talking about only real, valued polynomial or an X also would be, the variable X will also represent a real number only, okay? So this N has to be a non-negative integer, right? So you can't have a negative integer here. Either N can be zero or more than one, more than zero, but N cannot be 1.5 root two or negative one or negative 1.5, no. So that's the, you know, the speciality of polynomials. Every other thing is called algebraic expression. So there was a question board also, so where they are giving, they are giving and we'll see that question later on. So they're, sorry, what is this algebraic, yeah. Algebraic, algebraic functions or expressions. Now the word expression will get changed into function. The moment we go into 11th grade. So algebraic functions or in your language, algebraic expressions are all combinations of variables and coefficients, irrespective of the power on the variable. But if we are, the polynomial function is a special class of algebraic function where the powers of the variable has to be constant, non-negative, non-negative constant integers or whole numbers you can see, right? So X to the power Y, is it a polynomial? Yes or no? Is X to the power Y a polynomial? Yes or no? What do you think guys? X to the power Y is a polynomial or not? Hey, where is the answer? What happened to you? Polynomial if Y is a positive integer. If Y is positive integer, so X to the power Y is zero, then it's not a polynomial. That's what you're saying, Aryan. So in this state, it is not a polynomial because we do not know anything about Y, okay? So Y has to be a non-negative integer only then otherwise, what did I tell you? Non-negative constant integers, it has to be. So the variable, the power itself cannot be a variable. Okay, yes, we have to specify that Y is a non-negative constant integer, okay? So what about X to the power this Y one upon X? Is this a polynomial? X square plus one upon X is a polynomial. Yes or no? Is this a polynomial function? It's not a polynomial function, simply because this power of X is negative here. So not a polynomial function. And what about let me say this one? Sorry, so many people are there. X square into X minus one by X square. Is this a polynomial? No, Aryan, whole numbers are again, so usually we deal with integers. Whole numbers are four again. No one talks about that. So whole numbers again, all these things have been taught to you so that you can comprehend easily. But in the world of mathematics, no one talks about whole numbers. Yeah, so they will talk in terms of integers. So positive integers, non-negative integers, so that there's one uniformity across and very good. Yes, those who understand whole numbers, you can definitely talk about that. But as I told you, conventionally, we avoid using such terms here. So hence, we will talk about integers, rational numbers, irrational numbers, complex numbers. Okay, now, so clearly you're saying X square X minus one upon X square is not a, is this a polynomial? Yes or no? So this is a polynomial, yes. So hence, you have to simplify first and then see whether something is polynomial or not, okay? So this is very much a polynomial. Why? Because this is X cube minus one. Fair enough. What is X equals to zero? No, so hence, if X equals to zero, then you are specifying the value of the variable. Then it is not, you know, anyway, so it's a constant polynomial, sadly. So whenever we have, we can have, you know, FX is equal to constant as well. Either FX equals to zero or FX equals to constant. So these are called constant polynomial or zero polynomial. Okay, so this is a zero polynomial. This is a constant polynomial. These are also there, right? One upon zero into zero is equal to zero. No, division by zero is not allowed. Okay, so, oh, you're talking in this case, is it? Is this case, this particular case? No, it's anyway simplified as X cube minus one, yeah. So first of all, simplify it. See, for that matter, anything can be written. For example, even five can be written as five into X by X. That doesn't mean we will avoid X equals to zero case, right? So let's say there is a polynomial Y is equal to five. So someone can say, hey, I can write five into X by X. So hence, X should not be zero, right? Right, you cannot say like that. So hence, you know, what I'm saying is you have to simplify and see, you know, what is the simplest form of expression will be taken as the polynomial. I hope I made myself clear. Okay, now, so let's go do. So this is clear to you, A n, A n minus one. So don't get confused by the subscripting part of it. This has to be now very, very thorough with all of you. We are going to use this multiple number of times, especially in now upcoming grade. Y is one upon zero into zero, not equal to zero because one upon zero is not defined right here. So, you know, so you cannot multiply any not defined number with zero and say that it is zero. What are you multiplying with? So the multiplicands have to be defined before you multiply, you understand, right now. So you have to define, so multiplication is an operator operated upon two multiplicands, right? So the left-hand side and right-hand side multiply operator has to be defined first before the execution of the multiplication operation, right? So you can't really operate on something which is not defined. Is that okay? Yeah, so let's go do, yeah. So this is polynomial in variable x. There are polynomial where, for example, later on in your 11th grade, you are now going to study about circles, where you talk about these expressions. So x square plus y square is equal to one. So this is a polynomial, right? Or this is equation, by the way, but let's say I am saying x square plus y square minus one. Okay, so this is a polynomial in x and y, right? So there could be a polynomial in n number of variables later on and n, we are in the realm of algebra. So only thing you have to make sure is that power should be non-negative integers, that's it. Okay, let's go to this. Degree of a polynomial is an important concept. You are going to deal with this again and again, we calculus or algebra later on. So it must be very, very thorough. Exponent of the highest degree term in a polynomial is known as its degree. Highest degree term. So any polynomial will be made up of several terms. You have to identify though that term, which has highest possible degree, right? So in one variable, it is typically very easy to find out why because you can just see which number is the largest and pick that one up and you can say that's the degree. But let's say for a multiple variable, we'll see that as well. Though that is not there in your syllabus directly. So a polynomial of degree zero is called a constant polynomial, right? So for example, y is equal to seven is a constant polynomial y because it can always be written as x to the power of zero. So there is no, so this is called a constant polynomial. And what is our degree? Degree is zero. Degree is zero. But the point to be taken care of here is y is equal to fx is equal to zero is our constant polynomial, but its degree is not defined. Its degree is not defined. I hope everybody's aware of this. So a zero polynomial doesn't have a degree defined, okay? Now, polynomial of degree one, two, three is called linear and then what linear, quadratic, cubic, micro-derratic, quintic, so on and so forth. Naming again is not that important. You must just be aware of what exactly is happening. Fair enough. Any confusion, any questions so far? Sir, I had a doubt in a question related to polynomial. Yes. Who is this? Who's speaking? Sir, Meghna. Yeah, Meghna, tell me. Sir, I send it to you personally. Okay, on WhatsApp? Yes, sir. Okay, I will take that up. Wait, wait, we'll finish this and then we'll take it up, okay? I didn't get it, but when... Sir, I send it to you. Your number ends with five zero or something, no? I didn't get it. Maybe it is in the cyberspace right now. Sir, no, I sent it to you last Saturday, actually. Oh, my God. But your number ends with two zero seven six what? Number ends with? Wait, wait, one second, one second. I forgot your own number, is it? Zero five six, it ends with zero five. Zero five six, okay, it happens in your mind. So yeah, I got it. What is it? Oh, there is some figure and all that. Okay, we'll take it up as we discussed, no problem. It represents a what polynomial, okay? So there's a curve which is not that clear, okay? This appears, is it going down the x-axis also? I can't see it. It appears to be a cubic polynomial in this case. There are two... But it's not visible at how many points it's touching the x-axis, right? See, it may not be, see, I'll tell you. Another way of looking at it is how many turns the curve is taking. So usually, if there are two turns, the curve is taking and extending to infinity is negative infinity, extending to, I will show you in confidence thing. Just to give me two minutes, I'll show you. Yeah, all right, sir. I'll express it, don't worry. I'll tell you how to deal with such problems also. Okay, fair enough. So let's move ahead. So what Meghna is asking, I'll just take that question. So here is that, you know, let me just take it, keep. Okay, so here is what she is trying to understand. They go. So let me define a, let's take first four variables, A, B, C, and D. Okay. Now let me define, all of you are able to see this, right? This graph is visible to all of you, confirm, yeah. So let us say we have this curve, A x to the power three plus Meghna, I think this is exactly the figure which they have given to you, is it? Yes, sir, this is it. See, can you see that? So this is what I was saying. Anyway, let me complete this and I'll come to this. A x cube plus B x to the power two plus C x. I am taking a cubic, they go, right? It is touching how many times, only once. But can you see there is two turn in the graph. Now what they had given you is this, let me convert it to zero, this one. Let me convert this also to zero, this one. And this also to zero is what they have given you. So now it has only one root because it is touching the x-axis at exactly at one point. That doesn't mean that it cannot be a cubic one. See, the thing is any, this is called fundamental theorem of algebra. So any polynomial of degree n will have three roots. It has to have three roots. Whether it is real or complex, that's a secondary issue. But any polynomial of degree n will have how many roots? N roots, it has to have. So in this case, this particular curve is having, this is a cubic expression definitely because as you can see in the expression itself. So here there are one real root and two imaginary roots. So the way we had in quadratic, two imaginary roots. Similarly, in this case, there are two imaginary and one real root and one real root is zero. It's equals to zero. You can see that. So how to decide whether it is irrespective of the fact whether the polynomial or the curve is cut in the x-axis or not. See the number of turns, how many times it is swinging. So hence here, it takes, it goes up, becomes zero. So this is one turn and then again second turn, okay? So a cube a n degree, n degree, this thing, you know? Polynomial will have n minus one turns at max. So by the appearance and the shape of the curve itself, you can, see? So this is a polynomial of degree. Now if I change D, now here is what you would have got three roots, three real roots. Here only one, right? So either it will have three roots or it will have three real roots or one real roots. The other two will become imaginary. So can you see that? So this is how, this is what it is given. So hence by just by, let's say if this, this particular thing is given to you, this one. So if you don't know this curve, can you predict which kind of a polynomial is this? Is this, is a cubic polynomial, do you get the point? Irrespective of fact that it is not cutting the x-axis twice. Always remember it is the maximum number of roots is the number of, or it's equal to the degree. It need not be all the time. Is that understood Meghana? No sir, I'm not getting it. So what I'm saying is once again, let's say a degree three polynomial, how many roots it should have? Three. What type of roots? Real. Not necessary, that's what I'm saying. Understood? So if there is a degree three polynomial, let me reduce it to degree two. For a degree two polynomial, that is a quadratic polynomial, how many zeros will there be? For a quadratic polynomial, how many zeros do you expect? Okay, come on. How many zeros are there? Two, right? At most two. At most two, when you say at most two, here is where the difference between 10th and 11th grade lines. Now, every polynomial will have roots, right? A quadratic polynomial will have two zeros always. Understood? Always. At most two real roots is the right answer. Did you differentiate, did you get the difference between these two? Meghana again I'm asking. Once again, let me go to a note. Okay, there you go. So let us say we have y is equal to ax square plus bx plus. See, this is a quadratic polynomial. How many roots possible? Number of roots, two. Number of roots possible is equal to two, but a number of real roots possible. You will say at max two, right? Understood? Yes, sir. This is what, so let's say now this is ax cube plus bx square plus cx plus v. I should not be saying the roots, I should be saying zeros. It's a polynomial, so zeros. Now tell me for this cubic, how many zeros possible? Zeroes possible? Three zeros. Zeroes, there always be three zeros. There has to be. No one can deny the right to this cubic polynomial of having three zeros. Yes, what nature of roots will be, that's zero will be, that's something yet another fact, another thing to be discussed. So three zeros are there. Now in a cubic, and the other important thing you will be learning in now theory of equations and other things later on, that if all the coefficients are real, a, b, c, d, if all are real, then this can have even number of complex roots, right? Even number, meaning there cannot be one complex root and two real roots now. It has to be two complex and one. So what are the combination of all three real? This is one, all three real roots. What is the example of this? This is the example, which one? Let me show you. So in this case, here, can you see how many times it is cutting the x-axis? Trice. Trice, so this is this one, this one, and this one. These three are real roots, okay? But if I change the polynomial by increasing the value of d, the coefficient, the constant term, let's see. Now how many times it is intersecting the x-axis? Just once. Just one, where is the b and c? Gone, it has become imaginary, understood? So the imaginary roots are always in pair. Can you see? Right now? Yes, sir, got it. Always in pair. So b and c together disappear. It's not that it has a and b and c disappear, it's no. Together two roots will disappear, right? So all three real roots, this is one combination. Second combination will be one real and two complex. Okay, this we will study now. You are now getting the initiation into this all. You will be studying all of this now. So one real and two complex. It will never have this two real and one complex. Never possible. This is what is fundamental law of algebra. This is the first thing you're going to study and you're going to move it into grade 11. Basic thing. Okay, let's see. So can you explain why it can't have two real and one complex? See, complex roots have to be, again, you have to go into the proof of this, Aryan. Whenever there are rational coefficients or real coefficients, forget about rational. So hence in that case, you will, the roots, the roots which are there, if at all there is a complex root, it will occur in conjugate pair. So for example, if two plus three i is a root, two minus three i will also be a root. It has to be because you will see again, when we go into the proof of it, it will take some time, but this is, you will come out as an outcome, it will come if the coefficients are real, the complex roots will have to be in pairs. Is that understood? So hence, if there is one possibility of complex root, then it has to be a, you know, the other root has to be a, for example, do you remember if let's say, if all these coefficients in your ninth grade we studied, all of these are rational coefficients, then if there is a irrational root, it will appear in pairs. For example, two minus root three is one root, then two plus root three will also be a root of the same thing. These are called conjugate roots, right? So hence, you will never see only one, if the coefficients are rational, then you will never see one irrational root. It has to be in pairs. Similarly, if the coefficients are real, you will never see one complex root. It has to be in pairs. So if the complex roots are paired, then out of three, two are either zero complex root or two complex roots will be there. So only one is left for being real, right? Similarly, let's say there are the degrees five. How many real roots possible? Five real, zero complex. Then can this be four real one complex? Not possible. Then we have three real plus two complex possible. Four real and one complex, not possible. Five real and sorry, this is, sorry, pull down. Two real and three complex, not possible. One real and four complex possible. Did you get the point? And geometrically speaking also, see, when this curve, see what is the zero of a polynomial when the polynomial cuts the x-axis? Now there is a natural curve, right? It is turning like this. Now when it lifts up from the x-axis, exactly two points disappear. Do you get the point? Are you able to follow what I'm saying? Yes, sir, but what if it just touches the x-axis? Sir, when there was only point B. There are two equal real roots and one not equal real root. Understood? You are talking about this case, correct? Yes, sir. Your B and C are coincident. It doesn't mean that the roots have been disappeared or that the roots have disappeared. The B and C are sitting on top of each other. It's like when we say x minus two whole square is zero, how many roots are there? Two equal roots. Two equal roots. So here also what is happening? Out of three real roots, two are equal and one is different from those two equal. Understood? Yes, sir. But here also there are three roots. Never ever it will happen that one root is disappearing. Totally. Understood? Yes, sir. So this will be the case. Hence either one or three real roots or five real roots or seven real roots if the powers are odd like that. Understood? Degree is odd rather. Sir, but with the concept that we've been taught, if we look at it for the first time, we see that it intersects the x-axis at two points. That will give you the, see, that will, okay, what if I have this, tell me. They command low in the board paper or any paper for that matter. You have this equation, this curve, this one. How many roots are there? Oh, sorry. The question is what kind of a curve it is? What kind of polynomial it is? Is it linear? No. Certainly not because it doesn't appear to be a line. Right? So that means it is a polynomial of higher degree. Yes. More than one, right? Can it be three? Can this be three? It will never be three. Why? Because a polynomial of degree three will have at least one real root. See, we just proved. At least one real root has to be there. Understood? Yes or no? You tell me. Yes, sir. Yes, sir. No regular root. It has to be a, if it is a polynomial of degree three, it will cut the x-axis come what may, at least once. Got it? Okay, okay, sir. Yes, any odd, for example, if it is line, let's say, until it is parallel to x-axis, if a line is like this, y is equal to x plus one, how many roots are there or how many zeros are there for this? At least one is definitely there. That's equals to minus one, isn't it? So any, always remember, any polynomial of odd degree will cut the x-axis at least once. For all odd degree, I'm telling you, three, five, seven, nine, 11, all of these polynomials will cut the x-axis at least once. If a degree four is there, you can't say. Why? Because the polynomial would be of this shape. Something like that, w-shaped. So it is not cutting the x-axis at all. So come back to this and, so we can say polynomial of degree n will have n minus one turns, yes. They will have n minus one turn for any n degree, right? So let's say for cubic quadratic, it has to be one turn for cubic. It has to be two turns for bi-quadratic. It has to be three turns and things like that. Okay, so let's move on. So now, degree of polynomial is clear. This is our table to understand, to give you the brief of what is constant, what is linear, quadratic, cubic, bi-quadratic. Okay, no explanation required here. Degree four, degree three, degree two, degree one, degree zero, right? Now let's go to value of a polynomial. What is the value of a polynomial? So many questions on this. Oh, sorry. Yeah, so if fx is a polynomial and alpha is any real number, then the real number obtained by replacing x by alpha in fx is known as the value of fx at x equals to alpha and it is denoted by f alpha. So hence, so you have to just replace the variable by the value and calculate the, you know, it's like a try or treat it like a formula and then try to calculate the value. You will get the value of the polynomial clear. Okay, so you have been doing such kind of problems before also. Okay, let's move on. Now, a real number alpha is a zero for polynomial fx if f alpha is equal to zero. Example, if fx is two x square minus five x plus two then x equals to two and x equals to half are the zeros of fx. And f two is equal to zero and f half is also zero. So what is zero? So always remember, there's a mistake people do that they think that the number value itself becomes zero. No, so what is this? x equals to two and x equals to half are the zeros of fx. Two and half are not zeros themselves. So please be careful about this. So zero is a value of the variable which reduces the polynomial to zero. Okay, and a polynomial of degree n can have at most n real zeros, you can see I have n real zeros. So it can have n zeros. It is definitely having n zeros, but real as far as real zeros are concerned, n real zeros are there. Next. Now this is what we were talking about and this was a question you're bored as well. Geometrically the zeros of a polynomial fx are the x coordinates of the points where the graph y is equal to fx intersects the x axis, right? But that doesn't mean that if it is not intersecting it will not have zero, it will definitely have zero. Whether it is real or not that depends on whether the graph actually physically cuts the x axis or not. So here we can see this is, right? Okay, now in the adjoining graph points a, b and c are the zeros of the polynomial, this one. So I have shown this polynomial and there are a, b and c, three values where they are intersecting. So three real zeros in this case. Now these are important. The questions will be based on this sum and product of zeros. So if there is a polynomial quadratic ax squared plus vx plus c sum of zeros will be alpha plus beta, which is negative of coefficient of x divided by coefficient of x squared. And product of zeros is alpha beta constant term divided by coefficient of x squared. So always remember for the sum there is a negative sign and for the product there is no negative sign, right? So minus b by a and when you are calculating b the b will have the sign of, it's original sign. For example, if you're having x square minus two x plus three then b is minus two, not simply two. So here lots of people make mistakes. Keep that in mind. So here alpha plus beta will be equal to two simply and alpha beta is equal to three in this case, right? Similarly for bicodra, sorry, cubic. So there are three things here, alpha plus beta plus gamma minus b by a, which is coefficient of x squared divided by coefficient of x cubed and negative sign. Then again positive c by a, but this time around we say that sum of the two roots taken at two at a time. Alpha beta plus beta gamma plus gamma alpha coefficient of x divided by coefficient of x cubed and product alpha beta gamma is minus b by a constant term divided by coefficient of x cubed. Is that okay? So this is what is the theory part of it. Unfortunately, this is removed, but you must know this that fx is equal to gx into qx plus rx. It is similar to what our Euclid's division lemma. So Euclid's division lemma was on integers. Here we are talking about polynomials, right? Now what are these? What are these? fx equals gx plus, gx into qx plus rx where the degree of qx will be less than degree of fx for sure. And degree of rx is definitely either zero or less than gx, the divisor. But this has been taken away from your curriculum. So the problem will not be from this. So now let's enter into this game. So start solving. Tell me what will be the sum of the zeros of the quadratic polynomial px square minus kx plus six. Oh, sorry, it's given. Find the value of k. Sum of zeros is given. Find the value of k. Now let's apply and start solving. Yeah, guys, you're there. Hello, am I audible? Yes, quick. Okay, good. So some of the zeros of the quadratic polynomial. What is sum of the zeros alpha plus beta is equal to minus b by a. So minus of minus k by three, right? It is given as three. So k clearly is equal to nine. Good. This one. Find a quadratic polynomial with zeros are five minus three root two and five plus three root two. Okay. So any quadratic polynomial, I'll give you the basics first. So if you see any quadratic polynomial will be x square minus sum of zeros times x plus product of zero is equal to, sorry. This is, sorry. If you have to write the equation, this will be the equation. Otherwise, this is the polynomial. x square minus nx plus seven, right? So sum of zeros. So x square minus sum of zeros. What is this? Five minus three root two plus five plus three root two, x. And plus product of zero. So five minus three root two into five plus three root two. Right? So this is fx. Now there is a class of such quadratic polynomial. If you multiply this with any constant also, you are going to get the same zeros. So this is x square minus 10x. Okay. And what is this? Five square minus 18. So 25 minus 18, seven. So this is x square minus. This is one marker. Oh, sorry. This is two marks. Yeah. This is in sample paper. Next. What is that? Case study problem. So here is a, you can see the photograph. Application of parabolas, highways, overpasses, underpasses. A highway underpass is parabolic in shape. Okay. Now shape of cross. So this is just a, you know, lots of information given. Whether it is useful or not, you will see yourself see. A parabola is in the graph. That results from px is equal to x square plus bx plus c. Parabolas are symmetric about a vertical line known as axis of symmetry. The axis of symmetry runs through the maximum or minimum point of the parabola, which is called the vertex. It's called the vertex. I think this is the word. Okay. Now, so this is how it appears. So this is where a is greater than zero. This is where a is less than zero. Right? So you know that in the highway, overpass is represented by x square minus two x minus one. Eight, then it's zeros are, find the zeros. Simple. Find the zeros. Find the zeros. So a four and minus two. Right? So many people are saying four and minus two, very good. So you can find the zero directly. And, you know, since it is an MCQ, you can pick this one directly. Why? Because four plus minus two, some of the roots is minus two and product of loop is minus eight. So anyways, or you can solve it. How to solve? Split the middle term. So x square minus four x plus two x minus eight is equal to zero. So x times x minus four plus two times x minus four is equal to zero. So x minus four times x plus two is equal to zero. So either x equals to four or x equals to minus two. Chali, next. The highway overpass is represented graphically. Zeroes of a polynomial can be expressed graphically. Number of zeros of polynomial is equal to number of points where the graph of polynomial. So this is very easy. Intersects the x-axis. So see, irrespective of this diagram or not, if the diagram was not given, still you will be able to solve the question. So hence, in the case study based questions, I don't know whether this is actual quality question. But I don't see any relevance of the diagram here. Anyways, so let's go. You are not trying to find out anything from this diagram, so to say, in these two questions at least. OK. Now graph of a quadratic polynomial is a simple graph of a quadratic polynomial is a parabola. No problem in this, right? Representation of highway underpass, whose one zero is six and sum of the zero is six. Zero is zero. So representation of highway underpass, whose one zero is six and the sum of zeros. So alpha plus beta is zero given. And alpha is six, let's say. So six plus beta is equal to zero. So beta is equal to minus six. So hence, you know the, so it will be nothing but x square minus 36, right? Good. Number of zeros of polynomial this can have is the number of zeros that polynomial this can have is. Now this is a wrong question in my point of view. This is a wrong question. Why is it a wrong question? But yes, options are there. Now this is a little ambiguous in case of grade 10. So you might get if they do not mean real zeros, right? So hence, the number of real zeros is. So number of, in my opinion, when you write such kind of, when you attempt such questions, if you are finding some ambiguity, write that. So number of real zeros at max two. Number of zeros always two, right? So that you make your point clear. OK, so since it is asking number of zeros, so it will always be two. But if it is real zeros, then zero is also, you know, oh, sorry, number of zeros that polynomial can have is. Oh, anyways, sorry, this is not that. Yeah, this is straightforward. Not much of a problem. So number of zeros that polynomial can have is, yes, this is anyways clear. So they're not asking about real. Even if it is not mentioned, that's fine. Is it clear to you? What I'm saying is it is very much clear. The number of zeros that polynomial effects can have is two only. There is no ambiguity in this, sorry, right? Depending upon whether it is real or not, you can have zero, one or two rules. That's something else. Never mind. So you write two only. Good, so two. OK, this is actual board paper, 19, 20. One marker. Solve it. If one of the zeros of the quadratic polynomial is two, then the value of k is similar to what you can see. All right, this type of question you can predict now. There will be one question like that. Minus 10, number, if one of the zeros of the quadratic polynomial is two, then the value of, so, if fx is x square plus 3x plus k, so f of two will be zero. So f of two is zero. So that means two square plus three to two plus k is equal to zero. That means four plus six. So k is equal to minus 10. B, next. The quadratic polynomial, the sum of whose zeros is minus five and their product is six is a. Perfect. End out in this. So y, you can always do that. x square plus, sorry, minus sum of roots or zeros. x plus product of zeros. So this is x square minus, minus five x plus six. This x square plus five x plus six. OK, next. Find a quadratic polynomial whose zeros are reciprocals of the zeros of polynomial. This will take time. So three marker, take time and solve. So read the question carefully. What are the instructions and then do. And let me know the, when you're done, let me know. That's done. C x square plus B x plus one. And someone is saying A. A or one? One. OK, why? Find a quadratic polynomial whose zeros are reciprocals of the zeros of the polynomial, fx equals to this. So find a quadratic polynomial whose zeros are reciprocals. So let us say that the zeros of fx is equal to A x square plus B x plus C. These zeros are alpha and beta. And you have to find out this thing of one by alpha and one by beta. So basically your gx should be equal to x square minus one by alpha plus one by beta x plus one by alpha, one by beta. OK, this is what you are going to do. OK, so this is nothing but this is equal to x square minus alpha plus beta by alpha beta x plus one by alpha beta. Which is x square minus. What is alpha plus beta? Alpha plus beta is minus B. And what is alpha beta? Sorry, minus B by A. And this one is C by A plus one upon C by A. So C by A. Isn't it? So hence, this will be something. x square minus plus B by C. This is x also. So B by C plus A by C. So this means this is equal to C x square plus B x plus A. x is also there, divided by C. OK, so find a quadratic whose reciprocals are this. So yeah, so C x square plus B x plus A is also fine. No, you have to have this only. C x square plus B x plus A by C. Why is not C there? In your case, divided by C is not missing. Why is it missing? Guys, hello? Any explanation? Those who are, why is C not there? Sreejani, yes, Suresh, can you unmute and say why is C not there? Sir, I got the sum of zeros as minus B by C and the product of zeros as A by C. So either can be x square plus B by C x plus A by C or C x square plus B x plus A. So by C has to be also there. No, you missed, all of you missed the by C part. So I just wanted to understand why was that missing? Yes, Arun, what happened? Sir, what I did was I multiplied this with C, the whole thing with C. So that canceled off the denominator for the second term and the third term. But why did you, because there is a clear cut relation between the roots, reciprocal roots are there. So if you miss out on C, maybe you are changing the roots itself. You understand the point? Oh, OK, sir. So don't miss out on C, guys. Hello? All of you understood? So you would have done alpha plus beta, which is minus B by A. That's fine. Then alpha beta, which is C by A, that's fine. Now you have to find out this sum of reciprocal roots, the polynomial to 0's are reciprocal roots of the 0's. The new 0's are 1 by alpha and 1 by beta. So hence, alpha plus beta by alpha beta plus 1 by alpha beta. So hence, I don't think you should be missing out on C. B minus plus B by, yes, it has to be like that. OK, take care of that. Yeah, next. If 4 is a 0 of the cubic polynomial, find its other 2 0's, 3 marks. Easy. Other 0's are 2 and OK, let's see. Fair enough. So let's see. If 4 is a 0 of the cubic polynomial, x cubed minus 3x square minus 10x plus 24, that means f of 4 must be 0. Or you can divide by x minus 4. So what did you do? Or you can go for alpha beta gamma thing, whichever. There are two, three ways of doing it. So let's say alpha beta gamma are 0's. So what will be alpha plus beta plus gamma plus 4, let's say. I'm taking alpha plus beta plus 4 is equal to 3. And alpha beta plus 4 beta plus 4 alpha is equal to minus 10. And alpha beta into 4 is equal to what? Minus 24, am I right? Minus plus minus, yes. So like that, you get some equations here. So alpha beta is equal to minus 1 from here. And from here, you'll get alpha, sorry, this is alpha plus beta. Alpha plus beta is equal to minus 1. This you will get alpha beta is equal to minus 6, correct? Check if it is fine. So alpha plus beta is minus 1, alpha beta is equal to minus 6. And from here, yeah, so this will be redundant. So you can find out alpha minus beta from here. Alpha minus beta is equal to, what is alpha minus beta? Under root alpha plus beta whole square. And minus 4 alpha beta, isn't it? So this is nothing but 1 and minus 4 into minus 6. So under root 5, so alpha minus beta would be plus minus 5, OK? So you add these two. So what will you get? 2 alpha is equal to minus 1. You can take either ways. Only the sign of alpha and beta will change. So you can take either plus 5 or minus 5. Both ways, you get the same answer. So 2 alpha minus 1 plus 5 is 4. So alpha is 2. And beta is equal to minus 1 and minus 5. So minus 6 by 2, minus 3, correct? So did anyone deploy some other mechanism? You could have done one more thing. But you could have divided by this. xq minus 3 x square plus, sorry, minus 10x plus 24, right? And you would have found out the coefficient to be this. So you divided by x minus 4, whichever way. So you'll find this x minus 2 and plus 3. So x square plus x minus 6 will be the coefficient. Sorry, the coefficient, right? Did you get this coefficient? I didn't. So once you get this question, then you just split the middle term and find out the factors. You will get the 0. There are two ways. So either you go by division algorithm route or clear. Tell me. Yes or no? Clear? Any difficulty? So there are two ways you can do. One is through the division algorithm route or through sum and product of root root. Next. Again, one marker. 0 of the polynomial x square minus 3x minus n times n plus 3r. So minus m, 0 of the polynomial x square minus 3x minus mm plus 3r. You're saying b because sum of the root is minus 3r. So sum of root alpha plus beta. So let us say if the options are not even. Then what will you do? Did you check the option and solve? How did you solve it? Did you guys, or options only? If options were not given, can you find the 0 of the polynomial? You can find the 0 of the polynomial, how? So x square minus 3 and m into n plus 3 is there anyways. So you can always, you know, split the middle term. So yeah, so it has to be. On what? So I have minus now. So m plus 3 minus m. x minus mm plus 3. So this is x square minus mx. I think so. No, m plus 3x minus m plus 3x. And then minus mx minus mm plus 3. So x common, x minus m plus 3. And here minus m common x, oh wait. Minus x common, yeah. So why it is coming as minus then? x common, x minus m plus 3 and minus m common. Where, where, where? The second step is plus mx. In the second step, it is plus mx. Why, because x, sorry, so mx minus m plus 3, done. So this is x minus m plus 3 times x plus m. So minus m and m plus 3 are the roots of the 0s. Good, yeah, this one. Teacher asked 10 of his students to write a polynomial in one variable on a paper and then to hand over the paper. A plus B and AB relation. OK, yes, so we are done or possible. So the following where the answer is given by the students. 2x plus 3, 3x square plus 7x plus 2. So what is the question? How many of our 10 are not polynomials? How many? How many are not polynomials? So not is the keyword. Now again, what they have done is, I think this is complete root 3x and not root 3 times x. So treat it as root 3x, OK? So this is 1, 2, 3. These are polynomials. This one is not. This one is 4. This one is 5. This one is not. This one is. This one is. This one is not. So 3 or not. How many of our 10 are quadratic polynomials? Quadratic. So this is one here. This is not. This is not. This is not only one. Yeah, one, right? So this is 3, 1. Good. Now, Meghana, you will be getting curve graphs like this, very, very clear graphs. So there will not be any ambiguity. The number of zeros of polynomial is 2, clearly 2. What are the polynomials, by the way? Sorry, what are the zeros? So if they ask you to find the factors. So two factors, can you give me two factors of this? I'm saying factors. You did not notice. I said factors. So the factors will be x plus 3n, x plus 1, right? OK, good. Now, do carefully again. Similar type of question, which we have done. But now, please pay extra caution. Find the value of k. If the sum of the squares of zeros of the quadratic polynomial fx is 40, so squares of zeros. So alpha squared plus beta squared is 40. This is given. What is alpha plus beta? 8. What is alpha beta? k. So alpha squared plus beta squared, you have to find out. So if you square this alpha plus beta whole square, whole square is equal to 64. That is alpha squared plus beta squared plus 2 alpha beta is equal to 64. So alpha squared plus beta squared is 40. Plus 2 alpha beta is 2k is equal to 64. So 2k is equal to 24. So k is equal to 12. Very good. 3 marks, right? This one. Again, one mark. The number of zeros for a polynomial px where graph y is equal to px is given in figure is? How many zeros are there? Four. Now, you've got the point. This is where it is getting little, you know? So what finally answer you will write if such thing happened? So since it is subjective paper, you can still manage to write your assumption. So in this case, since it is up, you know, you can see clearly this is a bico-dratic. Bico-dratic, right? Why it is bico-dratic? Same sign of two extremes. Both are negative. And the curve is taking how many terms? Three terms. One, two, three. So it has to be a bico-dratic curve. If it is a polynomial. So it is given that in this polynomial. So it has to be a bico-dratic polynomial. So hence, bico-dratic polynomial will have either four real zeros or two real, two complex or all four complex. But here, you can clearly see one real, one real is there. And the other is also touching here. So there has to be four real roots. So you can see, you can say four real roots. Out of which two are equal. This is the right explanation, right? This one. Other zeros are, okay. Did you go through the division root? Can you do it without division root? Anyone? What do you think, guys? Can we do it without division root? Did anyone try without division root? Tell me, yeah. Yes or no? Anyone tried without division? So yes, definitely you can do the division part and solve it. So you now know that. So square of a quadratic polynomial not necessarily. See, it need not be a square all the time. So fx is, can I write like this? Two times, which is the coefficient of highest coefficient. And sum and product, yes, you can do that, but there will be four equations for and all that. So what I would have done is this. Can I write this as x minus root three? x plus root three, x minus alpha, x minus beta. What do you think? Am I right? Yes, guys. Is this correct expression? Is this correct? If yes, then why? If not, then why not? And why have I taken two here? Here. Why have I taken two? Can I not drop these two? I cannot drop these two because the highest coefficient. So hence, if you see after multiplication of all two x, this x, this x, this x should be two x to the power four. Hence I have to take that two. So I have made sure that this is done. And then whatever is the alpha beta, that will be taken care of, that's not a problem. So this is what it is right now, isn't it? So either I can go for equating the coefficients. So what is this? If you look at it, x square minus three. And then x square minus alpha plus beta x plus alpha beta. And if I expand this two, this will be x to the power four. Then x square goes to this. So this is minus two. What is that? Two times x squared. Yeah, so either you do entire calculation or you don't need to do entire calculation also, but let's do the expansion. So two x square, this goes to this and then this goes to this. What will happen? This is two times alpha plus beta x cube. Then that one will be two times, oh, sorry, no. This will become a longer route actually, because you have to multiply this also, sorry. There will be a lot of terms here. So this is another approach, but you can, instead of calculating all of that, what you can do is, so all the constants put together, you'll get the value of alpha beta once, right? And let us say, what will be the constant term here in this expansion? If I have to find out the constant term, that will be two times minus three, times alpha beta, is it it? So then that is minus six alpha beta. This is a constant term. If you expand the entire thing, you'll get two times, again, two times this minus three, and this alpha beta, I will give you the constant term. This is the only constant term you'll get. So minus six into alpha beta, and this will be equal to this minus three c. So alpha beta is equal to half. This is what we would have got. And coefficient, if I expand this and find out the coefficient of x, so what will be the coefficient of x guys? Simply, this is becoming two into minus three into this item here, alpha plus beta. Do you agree? Is there any other mechanism by which you'll get x? There is none. Yes or no? So the coefficient of x, x can be obtained. How? Only by two into this minus three and into this term. There is no other way out. So I'm giving you another way of, you can always divide and simplify, that's fine. So this will give me, coefficient of x is, how much? In this case, it is minus nine. So I will get three. So alpha plus beta is equal to three by two. So alpha plus beta is three by two and alpha beta is half. So I did not divide it actually. So are you guys getting the same thing? The alpha is minus, yes, I think. This is what you have also got it, minus one, minus half. But then I'm getting plus, you're getting minus. Hmm, alpha beta is this, okay. Opt in the other zeros. So you are saying minus half, minus one and minus half. So that means, okay, two into minus three, into minus half. But on this side, where we've canceled the three, minus three, so we get minus one, that's fine. No, minus three, minus nine, that will be three. On the right-hand side also we have minus nine. Oh, okay, so I'm sorry. Show how come alpha beta is coming out to be three by two. So did you check, everyone is getting minus one and half, or one and half? Everyone is getting minus one and half. So where is the error? Tata x square minus three, that's fine. And x square minus alpha plus beta x plus alpha beta is also fine. So the coefficient of x is two into minus three into, oh, my dear friends. Yes, there is a minus sign here also, because this is minus alpha plus beta x. So it will be minus three. Thank you. So alpha beta is minus three by two, hence done. Yes, thank you. Thanks, RM. So hence, you know, this is why, what happens if you do a lot of kitschery, don't do that. Okay, but yes, the formal route, which people have done is, they have multiplied the factors, you'll get x square minus three, then divide this entire expression. This is what everybody has done. You have done this way, yes or no? And hence, you have to be very thorough while dividing. So two x square, so two x to the power of four, minus three to just six, sorry, this is three, no, not three. So three to just six x square. So yeah, not here, it will be minus six x square here. Correct, now subtract, you will get three x cube, you will get three x cube, then this is minus five plus six, so x square minus nine x minus three. Then again, what? Plus three x, so three x cube, and minus three, three's are nine x, right? So you'll be left with x square minus three, so it will be simply one, and hence, gone. So this is two x square plus three x plus one, then you factorize this. You will get x equals to minus one, x equals to minus one. Good, this one, again four marks. Without actually calculating the zeros, form a quadratic polynomial whose zeros are reciprocal of the zeros of the polynomial, this. Without actually calculating the zero, take time, two minutes easily, and then do. Okay, so everybody is getting the same answer, if we get x square minus two x minus five upon three. Without actually calculating the zeros, form a quadratic polynomial whose zeros are similar to the previous ones, so they have some liking for reciprocal roots. So let's say alpha and beta over, let us say alpha and beta are zeros of five x square plus two x minus three. So my dear friends, alpha plus beta is going to be two by five minus, and alpha beta is going to be minus three upon five. Now, hence, what will happen? What do you need to find out? So the quadratic polynomial required will be x square minus one upon alpha plus one upon beta times x plus one upon alpha beta. This is nothing but x square minus alpha plus beta by alpha beta x plus one upon alpha beta, which is nothing but x square minus alpha plus beta is minus two upon five divided by alpha beta is minus three upon five times x plus one upon alpha beta is minus three upon five, which is equal to x square minus two by three x. Minus five by three. Did you all get this part? So this is nothing but three x squared minus two x minus five by, right, okay, four marks. This is easy, quick. So many questions on sum and product or form of quadratic polynomial. Yes, yes, yes. Minus three and two. So sum of root minus one. Satyam, sum of root is little, the product of four zeros are, okay, the sum and product of the root sum. Yes, sum of sum is alpha plus beta minus three and alpha beta is two. Which can I name it for x square plus three x plus two. Cool. Oh, similar. Three, similar question is there. Only three and minus three has become root three minus root three is root five minus root five. Similar, so divide and find the another quadratic factor, factorize it, get the zeros, done, so quickly, okay, okay. So fair enough, minus half and so hence the polynomial px will be how much? So px is equal to x minus root five, x plus root five times, let's say qx, right, which is the quadratic. So this is x square minus five times qx. This is how you can write there also. So px is equal to this. So what is qx? qx is px upon x square minus five. So let's do this, x square minus five within brackets, open this up, two x to the power four minus x cube minus 11 x square plus five x plus five. And yeah, so now two x squared, you'll get two x squared here, two x to the power four then minus five x squared comes here, divide, subtract minus x cube and then minus 11, which is minus six x squared, right. And then next is what? Plus five x plus five, that's it. And then minus x only. So minus x cube and plus five x, so plus five x comes here. This goes, so this is minus six x square plus five, sorry. Plus five, right. So minus 11, minus five. So there is, yes, how did you get this? Everyone got the answer also. Did I make some mistake? X to two x square, two x four, that's gone. X cube minus five, oh, sorry, sorry, sorry, sorry. Mine, oh, this is 10, no, hi. That's what I was startled by this, that this is not getting answered. This is correct now? Yes, yes, yes, thanks, sorry. So I missed that. So minus 11 plus 10, so x only, yes. So x, now it's okay. Yes, so x square, okay. Now, and rest all is okay. So five x comes down, plus five, that's fine. And then minus x, yeah, that's cool. So this is gone. So now what minus one, minus x square plus five, zero. So this is the quadratic factor qx. So qx is equal to two x square minus x minus one, which is two x square minus two x plus x minus one, which is two x common x minus one plus one x minus one. So it is x minus one times two x plus one. So what are the roots or zeros? So zeros are x equals to one, x equals to minus half. Boom, okay. This one, three marks. Find the value of k such that polynomial x square minus k plus six x plus two k minus one has some of it, zero is equal to half of their product. I said just this thing, Anish is there, Anish? Anish, are you there? Yes, sir. Yeah, Anish, Pratvaj, in Kannada. Yeah, you asked this question, no? So I was thinking x cube minus one actually. So x cube minus one is equal to zero. Or if you have y is equal to x cube minus one. So this is where there will be one real and two imaginary roots, okay? But if it is y is equal to x minus one whole cube, then it has three equal roots. So for to answer Sharduli as well. So one, one, one. So it is possible. So that time I was considering this. So cubic can have, yes, all three equal roots are there. Cubic can also have three. But if this kind of equation is there, then in this case, I was assuming this because here the roots are one omega omega square. So omega and omega square are the complex zero. So yeah, so hence you can have equal roots for just a collection further. Cubic also, yes. Okay, yeah, guys, now go ahead. Find the value of k such that this is done seven. Sum of it zero is equal to half of their product. Find the value of k such that the polynomial has some of its zeros equal to half of their product. So what is, oh, so you have to first find out alpha plus beta, sum of the zeros. How much is it? K plus six, okay? And alpha beta, product of zero is two, two k minus one, right? Sum of roots is k plus six, product of roots is two, two k minus. And what is given? Sum of zero is equal to half of their product. So k plus six is equal to half of this will be two k minus one, right? So k is equal to seven, okay? Three marks, not worth it. Anyway, find the zeros of the quadratic polynomial and verify the relationship between the zeros and the coefficients. Find the zeros of the quadratic polynomial and verify the relationship. Verify the relationship meaning, what do you do? Sum of roots and product of roots, you have to show that it is minus B by E and C by. Zeros are, different answers, same. Is it verified? Okay, so PX, whatever, seven Y square minus 11 by three Y minus two. Everyone did, did this part. So one way of doing it, this will be, you know, take the common denominator. So this will be 21 Y square minus 11 Y minus two upon three. Now it will be easier, isn't it? Below, yep. So hence, you can write this as 21 Y square minus 11 Y. So 14 is 42. So you can write this as 14 Y plus three Y minus two. Divided by three, which is seven Y common. And you will have three Y minus two, okay? Plus one, three Y minus two, divided by three, which is equal to one upon three, three Y minus two, seven Y plus one, isn't it? So since the mind is, you know, can't we remove the three? No, so PX is this much. So PX may, this three has to be there, one upon three. PX, PX's identity will change if you remove that three. One by three, so PX is this much, correct? No. So now they're asking zero. So zeros are Y is equal to two upon three and Y is equal to minus one upon seven, okay? Sum of zeroes, two by three, minus one upon seven, which is equal to 21, 14 and minus three, minus 11 by 21. Is it? It is, right? So yeah, you can equate it to zero directly, no problem, right? Because I was doing PX, I was writing PX like this, so hence I cannot eliminate one by three. If you remove this one by three, then PX is different. Then you have a different polynomial altogether with the same zeros. But polynomial can be, I told you, right? There will be a family of polynomials with the same set of zeros. You keep on multiplying with some constant K, you'll get the same, right? So, but then different, all polynomials will be different, but the zeros of all of them will be same. What does it physically mean? So these are the two points where they're cutting. So there could be infinite number of polynomials with the same zeros, same zero, but they're all different polynomials. Okay, anyways, so minus 11 by 21 is verified by, because minus B by A is, hold on, sum of zero should be 11 by 21. What's happening? So 11 by 21 done, and product of product of zeros is how much two by three into minus one upon seven, it is minus two upon 21. Which is true because C is minus two by three by A, which is seven is equal to minus two by 21, verified. This one, oh, sorry. This was asked in last two last year board paper. X squared, three X squared minus five. Okay, done. So see, I'm doing in a different way altogether. I'll show you how. If you're able to understand, so tell me, okay, and this into this, into this, into this. Yeah, this will give you B only. Okay, so did you understand what I'm doing guys? They go, I express this given polynomial as three X squared minus five into sum QX, which is X squared plus BX plus C. The coefficient one is one here. Why? Because this three is anyways present here, here. So hence, if you multiply this and this, you will get this three X to the power four. So hence, this is the expression. So you have to basically find out the K value, okay? Completely divisible, there is no remainder that is. So one equation I'm getting is K, the constant will be equal to the constant on the right-hand side. Constant on the right-hand side will be minus five into C. So minus five C. All other terms will have X. So they cannot be constant. The constant term will be only the last term here, multiplied by the last term here. So minus five C will be equated to K. Now come to the coefficient of X. What is the coefficient of X here? 15. So I wrote 15. Now let us see where do I get coefficient of X only on the right-hand side. So X, just imagine if you have three X squared you are multiplying, we just hide some things. Yeah, so let us say first you are multiplying with three X squared term. Now if you multiply with this three X squared term with X squared, you'll get X to the power four. Here it will get X to the power three. Here it will get X to the power two. And then minus five X squared minus five VX. Here is where you get the X terms, right? I've written 15 is equal to minus five V. So you've got V is equal to minus three. Are you getting what I'm saying? This is easier as you know, it will be simpler to find out, okay, so V is equal to minus three. Did you get what did I, what did I do? All of you, yes or no? Nice, whether you are getting it or not? Hello? So this is, this will simplify this. Now let us take the coefficient of X squared. What is the coefficient of X squared in the left-hand side one? And how can I get X squared on the right-hand side? So one is three X squared multiplied by C. So I can write this as three C. And this minus five multiplied by this X squared. So I can write minus five. So this is how I got, right? Three C minus five. Now three, now from here I can get C. C is equal to six by three, two, right? So I got three. So the moment I got, the moment I got C is equal to two, what will be K minus 10? Anish, can we find roots of three X squared minus five? This has to be root of other two. So we can replace X, we don't understand. Root of three X squared minus five will be root of FX, yes. So Anish, and then you can, okay. So you can put, yeah, under root, what five by three in the value and make it zero and find K. You can do that. What Anish is saying is this. He's saying since three X squared minus five is what root three X minus root five times root three X plus root five. That means two zeros are known. X equals to root five by root three. And X is equal to minus root five by root three are the zeros of FX. So these are zeros of FX. Meaning what? If I substitute this value in FX, I should get FX to be zero. So what he has done probably, did you do this part this way? But it might attract some calculation. But anyway, so what we have done is this. This is what you mean, Anish. Root of five by three whole to the power of four minus nine root of five by three whole to the power three plus root of five by three whole to the power two plus 15 root of five by three plus X plus K is equal to zero and you solve for K. Is this what you're talking about? Yeah, so hence it will be little cumbersome in terms of calculation. Okay, so in exam try to avoid such and especially when there are radical signs involved. The third method which most of you would have done is you would have divided this expression by three X square minus five and you would have got one quadratic expression. I think so, isn't it? That's what you did. Others, what was your procedure? So I took this route, okay? Where I just simply wrote a few linear equations solved for it and got the value up, right? So mostly people would have divided it and once you divide, then what did you do? Then Anish, then what you can do is once you get the quadratic, no, but then also not. Yes, then Surya, what did you do? Division, found factors after that. Reminder equated to zero. So hence you, whatever is the remainder, finally you would have got it, you would have equated it to zero, correct? And you will get K. These are the three methods. One method which I have been using so far is divisional algorithm method. So you quickly get the value of, so without multiplying, dividing and all that. This is Anish's method where he is substituting the one of the zeros and equating it to zero. The entire polynomial will be zero. And third one is divide and put remainder to be equal to zero. Yeah, three methods. Now, there are a few questions left. So, do one thing and do one more and then we will wrap up the session. Do this. If two by three and minus three are the zeroes of the polynomial, then find the values of A and B. This is, this should be very, very easy. Three marks. So again equating the coefficient root you can adopt. So, yep. So did you adopt this method or how did you do? So again, constant B here will be equal to A times minus two by three times three. So A times minus two by three times three. This will be the constant term when you expand it. Oh, B by AC by A root, okay, fair enough. So B is equal to minus two A. B is equal to minus two O, okay? And as far as A is, so next is seven X. So seven is equal to coefficient of X. How do I get X? So AX multiplies by three, so it becomes three A. And minus two by three multiplies with this X. So it becomes minus two by three, okay? So three A and minus two or one more A actually. So A is out and X times, so X times three, so A is out. And this one times three, so let it be A only. So X, this one times three, so three and minus two by three times half. So this is okay. Seven is equal to three. I hope this is clear to you. Again, what I'm saying is if you expand this, the coefficients where X is there, so it will be AX into X into three, sorry, into three, this AX into three. Plus the other one is this minus two by three multiplied by this X. And then A is also there, so A into minus two by three X. These are the two X terms in the right hand side. So coefficient is three A common minus two by three. That's what I've written. And that will be equated to the coefficient on the left hand side, seven. So this is nine minus two, seven. So seven is equal to seven by three A, okay? So A clearly is three. So what's B? B is minus two A, minus six. This is one method. Another method is sum of root product of root. So sum of root two by three or sum of zeros minus three is equal to minus seven upon A. This would be easier. Adopt this only. So A directly is three from here. And then product of root. So two by three into minus three is equal to B by A, right? So B will be two A, which is minus two A, minus six. Adopt this one, this is better. Fair enough. So mostly again, two questions are left. So you can, this is again similar. So no need to do it once again. Same on the same lines. Opt in all the zeros of this, when two A minus two are two zeros, you know what to do. Again, either you equate or you divide, equate the coefficients or you divide by, so hence what are the factors here? Here the factors are x minus two, x plus two. That means x square minus four. So this is the divisor. Divide this entire polynomial by this. You'll get another quadratic expression, factorize it. And equate to zero, you will get the other two roots. This is the root for this one. Here again, x square minus two will be a factor of this polynomial, divide, get a quadratic expression and factorize, factorize and then find the other zeros. Just we have done so many times now. Okay, fair enough. And this is how it should be written, right? This is actual board answers transcript, actually. So you can see this is of the division algorithm, but it is now out of syllabus. So again, given px, then gx, they have written, to check if gx is a factor, so basically they have divided and they have got a remainder not equal to zero. So in such cases, usually you should adopt this that px is equal to gx into qx plus rx. So please do this calculation. It will take one more minute to validate whether you have got the correct answer or not. So this algorithm, please keep in mind. But now that it is removed, so I don't think there will be any question related to that. Okay, any doubt now whatsoever? So if I have not taken any doubt previously also, so all those confusions are clear whether a cubic polynomial will have equal roots or not, how many equal roots, not equal roots, real roots, complex roots and all of that is clear, right? Okay, folks, so just to reiterate what I was explaining, that time that if you have y is equal to x minus one q, so this will have three equal roots. All equal roots are unity. Okay, but if you have this expression xq minus one, then this guy, this particular expression has one real root and two complex roots. That's what I was talking about this, assuming this. So that was the thing. So hence there will be three equal roots, yes. So no denying the fact. In fact, for that matter, any expression y is equal to x minus a to the power n will have n equal roots if that's x equals to a. But if it is x to the power n minus a, if this is the case, then there will be few real, few complex and complex roots will always be in pairs. Okay, clear. Dosto? Bolo. Any doubt or whatever in the previous things also, if anything is there, let me know. You can explain right now. Bolo, any question to be asked here? So polynomial is a big, big, big chapter and at least in sample paper, a huge weightage is there. So there are, you can see the sample paper, what all questions are there? How many marks? This is one mark, three mark and four, seven marks are there. So there are seven marks weighted out of 20 in algebra where linear equation, quadratic equation, as well as sequence and series and this polynomial. Four topics are there in algebra. This has the highest weightage or in fact equal to linear equation. Now, there is no distinction made as such that polynomial will have a seven marks weightage. They talk about 20 marks algebra. So you can expect a mix of weightages. So hence, polynomials might have less weightage. The AP sequence and series might have more. So in totality, but 20 marks algebra questions will be there. And in all probability, the word problem should be from quadratic equations or linear equations. So hence the five marker mostly will be allotted to those. So you can rest assured that this will not have a five marker question. Yeah, any other question, any other doubt, anything? Please tell me. Nothing, then we'll wrap the show up here and let's continue the pace. We will finish it. And then as we have mentioned earlier, we will up the ante. So we will start up, start taking up higher weight stuff once we are thoroughly done. So that in the month of February, you are relaxed. You don't need to revise anything. You just go write your PT, secure good marks and then stay happy. Okay, see you guys. Thanks for your time. I hope it made some sense to you today. Bye-bye, take care. Bye-bye, take care. Please keep solving. Please keep yourself in the same groove. Do not be relaxed. But at the same time, don't overstress yourself as well. I got to know that a lot of people are really using sleep over all this, all of this. No need, you are good. You're going at the right pace. All of you will be more than 95% marks. So don't worry about 10th board and all that. So I'm telling you, and with this time and this many repetitions, all of you will secure more than 95%. Okay, hello. So don't worry, relax, enjoy, and yes, but only thing is do not be complacent. Don't be over relaxed so that, yeah. Sir, I have not slept properly. No, no, no, please sleep, please sleep, papa. Sleep is the biggest, you know, this thing. That will create a lot of disruption. So for God's sake, do some physical exercise and, you know, please sleep. And the best way of falling asleep is take a book while you go to the bed and try reading a few chapters every day. Or some, you know, some book which is not that, don't read thrillers during the night time, something which is very interesting, then you will not be sleeping again. So something which is boring, read that and then you will fall asleep. But please have proper sleep because I have experience in my life. Lack of proper sleep will take away all your good deeds. So hence, there is no point losing sleep. Okay, hello. Or if anyone has any other, any other thing, please feel free to reach out. Okay, so as I told you, there is nothing for which you should lose your sleep, not at least for the board exams. And as I told you with the, I have the historical background myself. So I keep on seeing the performance of people every year. So by that standard and with the extra time, if you are consistently, since you would get 95 plus minus here and there and if God is very graceful, more than that also. So hence, even if let's say you get 99 or 91 or 89, whatever, immediately next objective is there for you. So hence, don't lose sleep over that. There are bigger goals to achieve. So hence, relax and be ready. Start preparing for that longer, this thing you can do. And there is absolutely nothing which should be, what do you say, which should take away your sleep. So please sleep, do some skipping, some cycling. If you have pet, take that pet out for a walk or a stroll. And the physical activity is equally important as academic series. So hence, otherwise the brain will just be like totally, what do you say, drained out and tired. The neurons will not fire at the right time when you want them to fire. So please take my advice, do some 15 minutes exercise this that here and there every morning evening. Go out, don't stay at home in the evening time. I know it's a pandemic time, but still you can take a walk on the street next to your home or you can just do a 15, 20 minute cycling. If you have a skipping rope, do that. If you have yoga, you know yoga or something like that, do that. If you're going to martial arts, please do that. So practice that if you, you know, but do some physical exercise. If nothing else, go to your apartment, top floor, come back, then go again, come back like that. Do something, some physical exercise has to be there. Okay, so I hope that makes some sense to you. Hello here, bye bye, take care, good night, sleep well.