 Hello and welcome to another session on polynomials In this session, we are going to understand what a zero of a polynomial is Okay, so now you must be wondering zero of a polynomial. What does that even mean? Zero is zero. What is meant by zero of a polynomial? Yes, when I encountered this statement I was feeling the same, but don't worry We are going to explain what exactly this word zero of a polynomial mean in the last session if you recollect we studied something called value of a value of a polynomial, isn't it? value of a polynomial wherein we used to substitute the variable with a fixed value and calculated the the entire value of the given polynomial, right? So let us take example again and try and understand once again what was what what could be the possible values of a polynomial and Then it will be automatically understood what a zero of a polynomial is. So let's take an example first. So let's us say px is x squared minus 3x plus 2. Now, this is a second degree polynomial Right, it is a quadratic polynomial and we try to calculate its value at different values of x So let's make a small table and understand this properly. Okay, so what I'm going to do is I am going to take some values of x here and then I am going to find out value of px here. Okay So let us say x is zero when x is zero Then this polynomial will become zero square minus three times zero Plus two this is two right Next is let us take x is equal to one If I take x equals to one it becomes one square minus three into one Plus two and this is equal to zero Let's take x equals to two. So it becomes two squared Minus three into two plus two this becomes zero It's not necessary that we have to take only integral values That is the values of x to be integers is not necessary. You can take anything Any real number but for the sake of understanding, we are just taking the values as integers It could be point two point three point nine Minus two by five root three root seven anything you can take. Yep any value of x is so let me just Write it here x could be x is any real number. There is no restriction that it has to be an integer No restriction x could be any real number. Okay for the calculation sake, we are taking some easier No, you know number. So let's say here it will be three square minus three into two Sorry three into three. So and plus two and this will be again nine minus nine plus two two Right. Okay. Let's take another one. Let's take minus one this time negative number. Okay So when it is minus one, what will you see? You see minus one squared minus three times minus one and plus two Isn't it and this will be uh one and plus three four plus two six Right. So another one. Let's take minus two So if you take minus two you what will you get? You will get minus two square minus three times minus two plus two And this value is simply four plus six That is then plus two twelve Right. So if you notice uh for different values of x you are getting You know x uh px is equal to two zero zero two six and twelve out of these these are You know You know of some importance to me why because if x is equal to one or x is equal to two in these cases Px is equal to zero. That's something interesting, right? So, uh, there are some values of x where the value of the polynomial is getting reduced to zero And those values of x. So let me just now write values of values those values of x or let me write out as variable Because x is a variable. I have to generalize this so values of variable which which reduces Which reduces or converts or you know evaluate whichever word you want to reduce means, you know, it is bringing it down Which reduces the value the value value of polynomial polynomial To zero to zero Are called are called zeros of Zeros of polynomial So in this case If you see x is equal to one Is a zero x is equal to two Is a zero of px now don't think that i'm equating x to zero no i'm i'm i'm calling it a zero because it is this value of x which is Making px zero and hence i'm saying those values of variable which brings down the value of polynomial to zero Will be called zero of that polynomial. So they are not simply zero They are zeros of zero of a polynomial. So what is a zero of a polynomial? zero of a polynomial is nothing but Is the value of variable isn't it it's a value it's not zero Zero of a polynomial is not zero necessarily it could be zero it could not be zero right So zero value wise i'm saying so zero of a polynomial value wise may be zero for example If you see x square minus x is equal to zero is is let's say is the given expression x square minus x So if you see at x equals to zero This item this also becomes zero. So that means Zero is making it making this entire expression value zero right so zero is One of the zeros of this particular expression. I hope this is clear. So when I write z e r o I don't mean the numeral zero. I mean those values value of variable right variable which which which makes px zero Right, so it could be x equals to alpha beta gamma whatever a b c whatever right any value or one You know any i'm just giving some you know I'm saying x could be anything to make you know px zero that depends on the polynomial which you are using right So if you see again x cube minus x square plus two x let's say this is The polynomial if this is the polynomial again if you see x equals to zero P zero will also be zero right if x equals to zero px is zero that is p zero is zero Right for x equals to zero p of x is zero And hence I can write this as p of zero is zero You can check whether it is zero or not put zero everywhere zero zero zero it will become zero so that means I can say zero is a zero z e r o of p x is it so There could be many more zeros also that will anyways talk about so one polynomial Depending on its degree can have more zeros But for your understanding, please remember a degree one Degree one polynomials degree one polynomials um Will have one zero will have one zero one One real zero one real right we will see degree two polynomials will have maximum of Two real zero what does what do these statements mean? We'll see but i'm saying maximum of two real zeros And uh three polynomial degree three degree three polynomial will have Three real zeros maximum real zeros real zeros Right or if I have to just remove this word let this word be not there so that it becomes yeah So degree one polynomial will have one zero degree two polynomials will have two zeros Degree three polynomial will have three zeros and so on and so forth how and why we will see as the course progresses But right now, please understand if a degree of a polynomial is known Let's say five then the number of zeros possible is five Okay, if the degree is nine number of zeros is nine that is yeah, so I hope you understand this How is that happening? We'll see as the course this course progresses and then you will get to understand that later Okay, so in this session, what did we understand we understood the meaning of zero of a polynomial So what is a zero of a polynomial? It's that value of the variable again zero of a polynomial is not zero always Numeral zero always zero of a polynomial is that value of a polynomial. Sorry that value of a variable Zero of a polynomial is value of a variable, which reduces the value of the polynomial to equal to zero Right zero is what zero of a polynomial is what zero of a polynomial is value of the variable which Reduces the value of the polynomial equal to zero Okay, which brings down the value of the polynomial to Zero right so zero is first of all not necessarily numeral zero Point number two zero is then is the you know is those that value or that value of the variable zero of a valuable zero of a polynomial is That value of the variable which brings down the value of the polynomial To zero right please understand that it will become more clear as we solve more problems on this