 Hey folks, so welcome to this new session and new topic on polynomials so We are going to discuss What all polynomials are there and how polynomials are defined why are polynomials called so and different types of polynomials and Primarily, then we'll be Trying to visualize polynomials on graph paper and then we will be studying about Let's say something called zeros of a polynomial and the properties around zeros of the polynomial So in this series of sessions, we will be covering all these topics, which I just discussed So in this session, we are going to discuss what all polynomials are and how do we define them and what are different types of polynomials? So let's do a quick Overview on polynomials. So we'll actually start with the very basic Concept in algebra. So as you know, these things are studied under algebra okay, so The first song was first thing which would have learned in algebra is something called a term Is it so a term is of the form of a times x to the power? Let us say n Where you know this a is nothing, but this is called the what is this called? This is called coefficient isn't it coefficient of this term and x is the variable so it need not be x you can have any other alpha Alphabet instead of x or any Greek alphabet as well and n is called the exponent Exponent or power right? So this is usually comprises this usually comprises our Algebraic term guys now next in line was our definition of let's say expression so a combination of Multiple terms separated by let's say addition and subtraction operator That is called an expression. So example of an expression for and for that matter what is an example of a term So let's say 3x square is a term then root 2 y to the power half is also term Like that. Okay. Now expression will be nothing but combination of these terms So hence if we say 4x square minus 2 y plus 3z X let us say this is this is a this is an expression, right? More specifically, this is a this is an algebraic expression, right algebraic expression now Category of x algebraic expressions are called polynomials polynomials and we define polynomials by you know polynomials and And the usual way of denoting polynomials is we write a capital letter and Within brackets the variable in which the polynomial is so px is a polynomial okay polynomial in x and it is generally expressed like this a n x to the power n Plus a n minus 1 x to the power n minus 1 plus so on and so forth. Let us say a 1 x and Plus a 0 Okay, once again, so if you see there are multiple terms, isn't it all these are terms this one is term This was another term another term another term now where if you see a 1 a 2 a 3 and So on and so forth till a n minus 1 and a n now don't get confused by the subscripts Which I'm writing so 1 2 3 So I can't use let's say I don't have hundred odd Alphabets to use as a coefficient as to denote coefficient So hence we do a smart thing. So we just write a and simply write 1 2 3 as a Suscript so that you know how many how many are here actually it starts from a naught So if you see there are n plus 1 coefficients, right? So all these all these belong belong to the set of real numbers So hence all are what what do I mean? I mean all of these coefficients are they have to be real numbers Real numbers. There is something called complex numbers as well, which you will study later, but In this case, we are confining ourselves only to real number coefficients. Okay now x is a variable, you know And n is a n is a non-negative Non-negative that means zero positive or any positive integer n is a non-negative integer So that means it's basically whole number right n can be a any whole number. So this kind of expression whenever you will see They are called polynomials, okay, they are called polynomials now Example of a polynomial is so let us say px is Equal to now I'm writing some examples of polynomial px is equal to 3x square minus 2x plus 1 this is a Polynomial right another polynomial. Let us say capital F of x is equal to is equal to 5 x minus 24 x to the power 3 Minus 51 y now if it is in x we can't write why let us say x to the power 4. This is again a Polynomial another example, let us say a small gx you can write this like this as well. Let us say root x square minus 7 by 5 let us say x Right, this is another example So if you see the coefficients are real 7 by 5 minus 7 by 5 and root 2 both are real and x is a variable and the powers are All non-negative integers. So these are all polynomials. What are not polynomials not examples of? polynomials, so if you see what are not poly no mails What are not polynomial? So if you see they're not polynomials basically so any expression like x to the power half Minus y by z. These are all let us say this is a polynomial This is sorry. This is an expression But if you see the powers are half and z is z has to be raised to power minus one here So hence these are all this is not a polynomial not a poly no mere Similarly, let us say root 2 X to the power root 2 right x to the power root 2 minus. Let us say y root y These are also these are also not a polynomial Now mind you in in the above definition. We have used px So we have used a polynomial only in one variable that doesn't mean it cannot Polynomials cannot exist in multiple variables. Okay, so a multiple variable polynomial p x y could be Could be let us say a A n x to the power n plus let us say b n y to the power n Plus a n minus 1 x to the power n minus 1 plus b n minus 1 y to the power n minus 1 and so on and so forth and let us say it is a 1 x plus b 1 y Plus a 0 plus b 0 where you know that a 0 a 1 a 2 all are and till a n all belongs to the set of real number and b 1 b 2 so on and so forth till b n also belongs to the set of Real numbers so hence you can have polynomials in one variable polynomials in multiple variables, right? But in our this course. We are going to restrict ourselves to the study of polynomials of only one Variable right next comes a concept of degree of a polynomial degree of Degree of a polynomial So what is that degree of a polynomial? This particular concept is going to be used a lot Then we'll be dealing with factorization of polynomials and Finding zeros of a polynomial and things like that. So what is the degree of a polynomial? It's nothing, but the it is defined as the exponent the exponent the exponent of the of the highest exponent of the highest degree term highest degree term in a highest degree term in a In a polynomial in a polynomial, right? The exponent of the highest degree term in a polynomial Okay, what does it mean? So let us take an example. So for example if I have fx is equal to x cube Minus 3 x square plus 2x plus 7. Let us say so what is the highest degree term? So highest degree term is this so degree of this term will be the degree of the polynomial So hence we say degree of fx is nothing but 3 Okay, similarly, let us say I have g of e is given as 2 t square minus 3 t cube plus 40 to the power 7 right so the highest degree term in this is this and what is the degree of this term 7 so what we are right? degree of sorry degree degree of GT degree of GT is equal to how much seven degree of GT is equal to how much seven now Okay, now We might have let's say polynomials like this So fx let's say fx y is another polynomial and it is like x cube y minus 27 x square y square plus let us say 42 x y to the power 4 What is the degree of this polynomial? The degree of this polynomial is 5 y highest degree term So if you see what is the degree here 1 so this degree of this term is 4 Degree of again this one is 2 plus 2 which is 4 and degree here is 1 plus 4 which is 5 so highest degree is 5 so hence degree of this polynomial is 5 I Hope you understood What is meant by degree of a? polynomial right Now let we will be taking up the different types of polynomial basis their degree