 I welcome all my friends on this session on polynomials and we have been doing a lot of sessions on number theory geometry and many other topics in mathematics and other subjects in this series and you know the this current session and the subsequent sessions are dedicated towards polynomials understanding their behavior and especially how to factorize polynomials that's the intent that's the objective okay so we are going to begin with simple definitions related to polynomials we need to understand the etymology of polynomials why the name polynomial what are polynomials how are they different from other expressions in mathematics especially algebraic expressions and things like that we will also deal with terminologies related to polynomials for example what is a degree of a polynomial you know and then what is the value of a polynomial roots of a polynomial or zeros of a polynomial things like that so we are going to discuss all of them one by one before we begin our understanding you know gaining understanding in this subject matter we need to understand what the name polynomial mean and it is interesting to note that the word polynomials is made up of two words of different origin so the first you know if you break this word into two so this is poly and the other one is no meal so poly as many of you know poly means many many and this is of Greek origin right so this is of Greek origin many right means many and what does nominal mean so this is a Latin origin word okay and it means name okay it means name so multiple many of names something like that right now why and how it got connected with the algebra of you know multiple terms is interesting to be noted and I think it will be because there were you know the notations of unknowns like x or alpha beta gamma or y or whatever was given as a name to some unknown quantities right and now the combination of those unknown quantities were polynomials and hence probably the name polynomial right so once again what I was trying to tell you is that we have been giving some symbols or names to unknowns for example x is a symbol or name given to an unknown quantity which we are representing through this letter x similarly you know other quantities like let's say if I have to add another physical code you are any quantity for that matter y to it so these are names given to these unknown quantities right and we are working on a combination of these unknown quantities through their names so there are multiple such quantities multiple such values represented by different names and hence the word polynomial okay so fair enough this is just for our you know understanding as to why these names and basically what is the genesis of these names but what exactly is polynomial so it now we come to the definition part of the polynomial so polynomial is an algebraic expression okay so we have studied what is an algebraic expression in previous grades so algebraic expressions are nothing but combination of terms with plus and minus sign right so polynomials are algebraic expressions okay with non-negative non-negative powers on the variables on the variables right so any expression having variables with non-negative powers what are non-negative guys non-negative basically means whole numbers right in 0 1 2 3 so on and so forth our non-negative integers right so let me write non-negative integral powers is it so this is non-negative integral powers on the variables non- negative integers that is right so how do we express this so usually if you see the expression will be something like this and we say it is to be P and within brackets we say X so it is what does it mean what does it mean it means an expression in X P for polynomial so let's say we are saying polynomial in X and why is this to be specified categorically is because polynomials could be in any variable right it could be X Y T S whatever you know whatever name you want to put or whatever you know letter or symbol you want to put it could be in that symbol so P of X and it is given by this expression just look at it carefully it is given as a not plus a 1 X plus a 2 X squared plus a 3 X cubed plus dot dot dot so many other terms let's say a n minus 1 X to the power n minus 1 plus a n X to the power n okay this is what is the general expression right of a polynomial where where a not a 1 a 2 dot dot a n minus 1 a n which also you know as coefficients of the terms coefficients of terms right all the terms right are real numbers are real numbers okay are real numbers and n is a non-negative non-negative integer understood so don't get bogged down by so many you know terms and terminologies here so P of X so it's an expression statement in X which is of this form what form a not plus a 1 X plus a 2 X square plus a 3 X cube so on and so forth like that like that so let me take an example and tell you so for example I can say again P of X is 1 plus 2 X plus X square so this is a polynomial in X right where if you see this item is a not which is a constant term this is coefficient of X I'm writing it as a 1 and this is coefficient of 1 is the coefficient of X square here so a 2 okay so only three terms are there in this polynomial another example could be P X could be let's say root 2 real number no problem then X square right or rather let me just first have a you know X and then here it is 2 X square minus 3 by 2 X to the power 3 okay this is another polynomial in X again so this one will be a 0 here you know constant term this is a 1 this is a 2 and in this case minus 3 by 2 becomes a 3 right so these are polynomials example of polynomials okay so what are not polynomials then so if you see if I have an expression in X for example root 2 by X minus 1 if this is an X this is a this is definitely an algebraic expression but this is not a polynomial not a polynomial why is it why is it because if you look at it carefully it is root 2 X to the power minus 1 minus 1 correct and my dear friend this is not allowed negative it is a negative integer and negative integer is in polynomial powers not allowed right you see here only non negative integers are allowed but here it is negative so hence right also if you see 1 by 2 X to the power or root of X let's say 1 by 2 X root X minus X square this is also not a polynomial not a polynomial why is it why it is not a polynomial is because if you see this is can be written as half X to the power half minus X square now again this half is creating the trouble this is not allowed why because we have only integers allowed here right so this is not an integer not an integer so this is not a polynomial okay so non negative integer it has to be and only then it will be polynomial next question would be is a polynomial only in one variable so you see here it is only one variable X but we can have polynomial in multiple variables X and Y example let's say X square plus 2 XY plus Y square is a polynomial in two variables is it so this is polynomial polynomial in two variables right so we can have polynomials in many multiple variables so for example P X Y Z let's say this expression is something like X cube plus Y cube plus Z cube minus 3 X Y Z let's say okay so this is a polynomial again polynomial polynomial if you check all the powers are non-intake non-intake you know non-negative integers everywhere the powers are non-negative integers okay so polynomial in two variables this is polynomial in three variables is that okay polynomial in three variables so likewise likewise you can have polynomial in any number of variables so 50 variables 70 variables 80 variables 1 million variables depending on the case you can have there's no restriction on number of variables on a polynomial right but the only condition is the coefficients must be real numbers so for example here is one here it is one here it is one and here it is minus three so coefficients are real and power should be non-negative integers non negative integers this is the only criteria for defining a polynomial so if you see all polynomials all polynomials are algebraic expressions for sure algebraic expressions okay but all algebraic expression and I'll write in short all algebraic expressions need not or rather are not are not polynomials okay this you please understand right all algebraic expressions are not polynomial for example root X is not a polynomial not a polynomial is it but all polynomials have to be for example X is a polynomial it is an algebraic expression X square plus 2 algebraic expression X square plus X plus 2 algebraic expression as well as polynomial right so this is what is about definition of polynomials and you know that these independent or individual entities here these are terms what are they terms all terms are joined with plus or minus sign these are all terms guys the ones which I'm circling including the coefficient with their signs are terms example so let's say if I have 3x square minus 4x plus 2 so terms there are three terms three terms in this first term is 3x square then minus 4x then 2 these are the three terms in this polynomial right these are the terms now these numbers are called coefficients so minus 3 minus 4 are coefficients of the terms respective terms coefficients right so the word is coefficients okay so what is the polynomial guys is if you see now a polynomial is nothing but a combination of different terms such that the coefficients are real and the powers are non-negative integers on the variable non-negative integers on the variable so root 3x square is very much a polynomial root 3 is a real number that's allowed but you can't have root 3x like that this is not a polynomial not a polynomial okay but this one is definitely a polynomial so in on coefficient there is no restriction it could be any real number but on the powers of variable it must be must be again repeating must be non-negative integers only then we call them polynomials I hope this is understood in the next session we'll take up some more definitions around polynomials