 How folks so in this series of sessions, we have already discussed types of polynomials We have discussed polynomials in general what polynomials are how do we define polynomials and We have also defined or we discussed in the last session different types of polynomials We saw linear bicarbonate quadratic cubic and all that and In this session, we are going to discuss. What does it mean to or what does value of a polynomial mean? Okay, so you now know there is a polynomial. Let us say fx or px. Let us say there's a polynomial px Polynomial px. Let me yeah, let us say there is a polynomial Px so p of x is given as a n x n plus a n minus 1 x n minus 1 plus So on and so forth. Let us say a 2x squared plus a 1 x plus a naught So these are this is our polynomial. Now, what is value of a polynomial mean value of a polynomial means that At any given value of x, let us say at x equals to let us say alpha Right. So what if you if now someone says value of a polynomial at x equals to alpha is nothing But you replace what you replace Replace x or the variable the variable in this case x variable by the given number by the Given given value given value and that value here is alpha. Isn't it? So if you replace it So it will become p of so wherever you see x you replace it by alpha So it is p of alpha is equal to a n alpha n then a n minus 1 alpha n minus 1 Plus dot dot whatever it is. Let us say a 2 alpha square plus a 1 alpha plus a naught so this will be value of the Polynomial so I hope you would have understood but you know, I know we need to Do or let's just give you some more examples to understand this. So not to worry. Let us see What does it mean? What does it mean to find out value of a given polynomial? So let us take an example. Now. I have an example Px is let us say 3x square minus 2x plus 1 Okay, this is one of my so if you have studied the type of types of polynomials You would have guessed it by now that this is a quadratic polynomial quadratic polynomial Why because the highest degree term is this and the degree is 2 isn't it? So this is a quadratic polynomial and I want to find out value of Value of what Px so always value of the polynomial would be found out at let us say x equals to 0 Okay, so if I what does it mean? Then you do don't need to do anything you just write simply write P and replace x by 0 wherever you see x replace it by 0 So this 3 into 0 square minus 2 times 0 now this point which I am making is Into it's not it doesn't mean decimal. So it is a multiplication sign and then plus 1 So hence you'll see it is nothing but 1 so if someone asks you about the Value of the polynomial when x or the variable is 0, you know, you know What what will the value the value will be nothing but whatever is a constant term? Can you see only constant term is right? So let us now find for this same polynomial P1 What does it mean? It means replace x by 1 so 3 into 1 square minus 2 into 1 Plus 1 so which is nothing but 3 minus 2 plus 1 which is nothing but 2 so we say Px is equal to 2 at x equals 1 Okay, in the same example, let us say what do we get if I have to find out P of minus 2 That means find the value of the polynomial at P is equal to minus 2 So what will it be so you replace x by minus 2 wherever wherever it is so minus 2 square minus 2 times minus 2 Plus 1 which is equal to 3 into this is nothing but 3 times 4 This is not so let me write into only so minus 2 into so this will be 2 into 2 and then plus 1 So which is nothing but 12 plus 4 plus 1 so 17 Right, so this is how you you will calculate the value of a polynomial at a Given value of x or the variable similarly another polynomial. Let us say g t is equal to 3 t cube minus 2 t Square plus. Let us say t only okay three terms are there We know that and what type of polynomial it is if you see this is a cubic polynomial Isn't it because the highest degree term is 3? Cubic polynomial so hence if I put g of 0 What will I get I'll get 3 into 0 square sorry cube Minus 2 times 0 square plus 0 which is equal to 0 there is no constant term Then the moment you put 0 in the polynomial you'll get the value as 0 what will be g 1 G 1 will be 3 into 1 cube Minus 2 into 1 square plus 1 so what is it 3 minus 2 plus 1 again 3 minus 2 plus 1 that is 2 so g 1 is equal to 2 Okay, so value of the polynomial at x equals to 1 is nothing but 2 this is what is called Finding out the value of a polynomial now. Let us solve our question. Let us say I have a polynomial px And this is a quadratic polynomial quadratic polynomial Okay, polynomial and it is given that p 0 is equal to 1 P of 1 is equal to 2 and p of minus 1 is equal to let us say zero So you have to find out find the polynomial find the polynomial find the polynomial right so solution how do we solve it? So, let us say the polynomial px is nothing but ax squared plus bx plus c I know that our quadratic polynomial will always be of this form a b ax square bx plus c is it it and And mind you a cannot be zero if a a becomes zero Then what happens? This is reduced to a linear polynomial. So hence. This is my desired quadratic polynomial I have to just find out the value of a b and c and hence I will be able to find the polynomial So, let us see how to do it. So p of zero. Let us put x equals to zero So what will you get a times zero square plus b times zero plus c and it is given that p of zero is one So this value is one that means what c is equal to one. So you found out one value now P of one is given as two so p of one is equal to two that means you have to replace x by one all the time, right? Wherever x is so a times one square plus b times one plus c c I can write as one because we just found out c is one. So what do we get a plus b? Plus one is equal to two that means a plus b is equal to one Okay, now third is p of minus one is equal to zero that means what p of minus one will be a times minus one square plus b times minus one Plus c was one. So let us write one. So what do I get? So it is a minus b Plus one is equal to zero or a minus b is equal to minus one Isn't it so now if you see let us see let us say this is equation number one This is equation number two. So if you add one plus two, what will you get if you add? add one Plus two so adding equation means adding LHS to LHS RHS to RHS So you'll get two a isn't it because what is it? It is nothing but a plus b plus a minus b in the RHS So eventually bb will get cancelled. We get two a first and then on the RHS you will get zero Why one minus one which is zero? So hence a is coming out to be zero, which is not possible Isn't it so hence we will not not get a polynomial. We will not get a quadratic polynomial where a is A is zero. So hence let us tweak it a little bit. Let us say instead of saying that p minus one was zero P minus one was P minus one was zero. Let us say we are We have P P minus one is also equal to let us say two Just for the Understanding sake, okay, so let us say we change it because yeah, so let us say it is two So what what will happen? So you will get let us say now you equate this to P of minus one is equal to So I had written here zero actually P one was two Yeah, P minus one. Okay, so let's not keep it same. Let us write P one P minus one was three So instead of here, I use you have written by mistake two. So let me play it be Three, okay. Now what so a minus B plus one here. I use it correctly But then there I was writing to never mind so a minus B this a This a here minus B plus one is let us say three. So a minus B will be nothing but two Isn't it a minus B will be two now again add it add both. So what will you get two plus one and Here you will get one plus this two is nothing but not zero but three So hence my friend, what do I get a is equal to three by two, right? So if a is equal to three by two, I can find B why from one from from one A is equal to one minus B is it or sorry B is equal to one minus a B is equal to one minus a so hence B is equal to one minus three by two, which is minus half So hence what is my equation? Now we are sorry polynomial. I got all a B and C So polynomial P X will be equal to what is a Three by two X square. It is B minus half X plus C. That is one. Now. Let us check whether our Result is correct or not. So P zero should give you zero clearly P zero is giving you zero put X goes to zero You'll get P zero is zero P one should give us two. So let's put one So it is three by two minus one by two that is one and plus one is two so P one is also correct and P of what was the third one minus one, right? P of minus one must be three So let us check P of minus one is equal to three by two minus one square minus half times minus one plus one, which is nothing but three by two plus one by two That is two plus one that is three. So all the three values are satisfied So hence my polynomial is nothing but P X equals to three by two X square minus half X plus one Hope you understood what is the meaning of value of a polynomial and let us say if If the if some values of polynomial at different different points are given You can find out find out the polynomial itself And if you notice to find out a quadratic polynomial, you will require Three such values. So if there will be a cubic polynomial Then you will be requiring four such points to find out that cubic polynomial