 Hello friends and welcome again to another session on factorization of polynomials and we have studied about polynomials, what a polynomial is, what an algebraic expression is, different types of polynomials and other things in the previous sessions. Now in today's session we are going to have a look on what is meant by value of a polynomial. This is going to be very important in terms of your understanding of all what we would be discussing in algebra, in this grade, later grades and mathematics as a whole. Hence, please understand this concept very thoroughly. This is going to be used again in let's say later in chapter on functions and other things. Let's understand what a value of a polynomial means. Before that, let us write a polynomial and take some examples. First we see some examples and then we will try to generalize it. So I am writing example, example one. So let's start understanding through examples. So let's say a polynomial px is given and let us say this is x square minus 3x plus 2. So if you remember, whatever we discussed in the previous sessions, this is a quadratic polynomial. Order is, we learned how to find out order, so highest order term, sorry not order, I am really sorry, this is degree. So degree is degree. So we learned about degree. Degree is highest degree term out of all the terms present. So there are three terms. So highest order is 2, sorry highest degree is 2. So hence degree of this polynomial is 2. We are trying to find out value of this polynomial when, let's say given value of x. Let's say what happens when x is 0. Let's say if I want to find out the value of this polynomial at x equals to 0, it would mean what will be the value of px when you deploy 0 into this and wherever you see x put that 0, that is what is meant by value of polynomial at 0 and we write like this. So we write 0, p of 0. So value of this polynomial when x equals to 0 will be just replace or substitute x by the number which you are trying to calculate value on. So x square, so 0 square minus 3 times 0. Now this dot represents a multiplication sign plus 2. Now it is a simple arithmetic calculation and you will see this is equal to 2. So we say value of value of px at x equals 0 and it is represented by p, the name with or the letter or symbol by which we are representing the polynomial and we write 0 within this bracket and then we simply calculate or it's as if we are trying to use this formula to find out the value of p0 and this is right. So we say value of px at x equals to 0 is p0 and we say p0 is equal to, sorry, this is not 0, this is 2, my bad, so this is 2. So p0 is 2. Isn't it? Let's find out another value. So let's say at x equals to 1, px will be how much? So px, we write px equals to 1. Many times you will see written like this, px equals to 1 will be simply replace x by 1. So x square, so 1 square minus 3 times 1, 3 times 1 plus 2 and this all, this comes to be 0. So we say p1 value of polynomial px at x equals to 1 is 0. One more example, let's try to find out at x equals to 2. That means when value of x is 2, then what will be the value of the polynomial? So px equals to 2 is simply 2 square minus 3 into 2 plus 2 and this also happens to be 0. So we see that at x equals to 1 and x equals to 2. In both the cases, the polynomial value px is coming out to be 0. Let's check some more. So at x equals to 3, what will happen? So we will write px is equal to 3 is equal to 3 square minus 3 into 3. So wherever there is x, I have written 3 simply and this is 9 minus 9, so 2. So this is what is meant by value of a polynomial. Let's take another example. Let's say example 2. If fx now, the polynomial symbol is given by f of x is equal to, so we say this is a polynomial in variable x and it is given to be equal to 2x cube minus 13x squared plus 17x plus 12. Let's say this is a polynomial. What is the degree guys? So degree of fx, clearly, degree of fx clearly is 3. Highest degree term is x cube, 2x cube, so 3. Now the question is, find f2. So what is the value of polynomial at x equals to 2? So let's find out. It's simply f of 2 will be equal to 2 into 2 cubed minus 13 into 2 squared plus 17 into 2 plus 12. Is that so? This is how much? 2 into 8, correct? Minus 13 times 4 plus 17 into 2 is 34 plus 12. So if you calculate all of this, you will get a value of n. This is f2. Now let's say they are asking you to find out f0. Very easy. What is f0? That means put x equals to 0 wherever you see x. So what is it? 2 times 0 cubed minus 13 times 0 squared plus 17 times 0 plus 12. So simply 12. Is that it? This is what is meant by value of a polynomial. So generalizing we say value of polynomial at a value of polynomial. What is that polynomial whose value we are trying to find out? Value of polynomial p of x or fx or whatever is the name at x is equal to a. Let's say the value of polynomial you want to find out at x equals to a. In these cases, if you see a is equal to 2, in this case a was 0. So if I want to find out value of the polynomial px at x equals to a, it will be, it is, value of polynomial px at x equals to a is simply p of a. That means wherever you see x, replace that by a. It need not be a polynomial in x. It could be in tu, y, z, whatever. So whenever you want to find out the value of the polynomial, you need to insert or replace the variable by the value and then calculate the value of the entire expression or the polynomial. You will get the value of the polynomial at that value of the variable. So pa, so let's say if p of x was given by a n x to the power n plus a n minus 1, x to the power n minus 1 plus a n minus 2, x n minus 2, so on and so forth, a 3 x cube plus a 2 x squared plus a 1 x plus a naught, let's say. If px is this much, then pa is the value of a or value of polynomial at a is simply a times n plus a n minus 1, a times, right? So wherever you see x, you simply replace it by a. a n minus 2, a n minus 2, likewise. Finally, a cube, a 3, sorry, a cube plus a 2 a square plus a 1 times a plus a 0. Now, don't get confused between the coefficients a 1, a 2, a 3, a n and the variable value a. These are coefficient guys. These are coefficients and the suffixes represent, you know, just their number at, you know, because there are lots of coefficients. So hence, I have used a suffix system a naught, a 1, a 2, a 3, a like that, right? But this a here is the value of the variable. This one is value of the variable. So these are two different things. Please keep that in mind. I hope you understood what the value of a polynomial is. In the next session, let us learn what a 0 of a polynomial is, okay? Yeah.