 In this session we are going to discuss a very important aspect of polynomial and that's called degree of a polynomial. Now this particular aspect is very very you know important because there are lots of theories, theorems and results which will be based on degree of a polynomial. So it's very important to understand the essence of this particular concept okay. So once you are sound in the this particular concept you will be able to understand multiple other theorems and there will not be any confusion neither will you make any mistakes while solving different types of problems. So please pay attention and try and understand what exactly does degree of a polynomial mean. So in the previous session we saw the definition of a polynomial and we said any polynomial p of x which is let's say we are now right now talking about single variable polynomial and it was given by this particular expression wherein it was a n x n. So I am now writing in the reverse order a n minus 1 x n minus 1 in the previous session I had given you the reverse order of this expression but then doesn't really matter because either you write this way or the other way in which we discussed last session both are same. So a n x n plus a n minus 1 x n minus 1 and a n minus 2 x to the power n minus 2 so on and so forth. Let's say it goes like a 3 x cubed then a 2 x squared then a 1 x plus a naught right these are the terms right. So what did we understand last class these were the terms these independent entities here these are the terms is it they are all they are all terms terms of the given polynomial and what was the criteria criteria was that n has to be n is a non negative non negative negative integer it cannot have a value of a rational number or a negative integer right n is a non negative integer only it has to be only an integer all all powers right all powers because now in this case you see all powers are related to n so if n is integer so all other powers have to be integers as well so these are the powers powers on the variable okay powers on the variable have to be integers that has to be reemphasized again and again okay so here also if you see x to the power of zero so that's also a non negative integer isn't it so how do we define the degree of a polynomial guys so we define it like this the exponent the exponent the exponent the exponent of the highest of the highest highest degree term okay exponent of the highest degree term so there are many terms you have to identify that term which has the highest degree in a polynomial in a polynomial in a polynomial is known as is known as its degree so let's take a few examples to understand this well okay once again what is an x the degree the exponent of the highest degree term so first of all we have to identify the highest degree term and that degree becomes the or that power becomes the degree of the polynomial what do I mean let's start tx let's say is equal to x plus 1 right x plus 1 what is that degree what are how many terms first of all there are two terms in this clearly this is t1 and this is t2 t stands for term let's say so there are two terms out of these which term has the highest degree of the variable clearly the first term has the highest degree because here it is one and here it is x to the power of zero so this one is the highest degree term and what is the degree of this term two sorry one so degree of degree of px I am writing like that degree of px is all how much one correct let's take another example so let's say this time px is this is first example this is second example let's say in this case now px is equal to 2x square minus 3x plus 1 how many terms so these are three terms there are t1 t2 t3 right three terms and which term has the highest power on the variable clearly this term is the highest power of variable and what is that highest power 2 so hence degree of px is 2 isn't it degree of px is 2 then let's take another example to make it more clear let's say we have px is equal to 1 by 2 x to the power 7 minus 3x square plus 1 right so what is the degree in this case if you check there are three terms again the highest degree term is this so hence degree degree of px is equal to 7 correct now you must be wondering whether it is true only for polynomials of one variable or what happens if there are multiple variables so let's see this so p of let's say x and y two variable polynomial we'll be writing let's say it is x square y square minus x plus y okay how many terms again there are three terms three terms right three terms and which is the highest degree term clearly this is only one you're only one there are two and two right in this case this is the highest degree term but in this case if you see total degree it will be nothing but you know it it is as if two times x and two y's are multiplied together so in this case degree will be total power in on this term is how much two plus two four so degree of pxy in this case is four okay so what do we get we understand that in you know you have to find out the highest degree term and if there are multiple variables in that term then you have to simply add the powers on that term so y is degree four here if you look at it this is x into x into y into y right so there are four it appears to be that there are four variables multiplied together hence it is degrees four I hope this is clear let's take another example to make it more clear so let's say I have another one p x y z in three three variables and it is let's say x plus y plus z minus x y minus x z plus x y z okay so what will be the term or what will be the degree in this case so here it is one one one here it is one one so this is two order or a second degree term this is also two two are being multiplied but here there are three so in this case the highest degree term is this so degree of degree of p x y comma z is equal to three correct I hope this is understood now let's take another one so that more the examples better is the clarity so x y z and let's say u four variables now here x plus u plus y z minus x square u plus x y z square so what is the degree here in this case the highest order term is simply this one higher degree term sorry and how many variables are being multiplied over there so what is the total degree so one one and two so total is four so in this case degree is nothing but degree of p x y z u let's say this was the polynomial in four variables is nothing but four okay so that's what we observe so degree of any polynomial is a non-negative integer that's one observation isn't it so degree is clearly degree of degree of polynomial of polynomial polynomial is is a non-negative non-negative integer this is very clear non-negative integer it could be one two three four five six anything you know depending upon the number of or the power on the highest order highest degree term right now you must be wondering can there be degree zero polynomial yes so degree zero degree zero polynomial degree zero polynomial what is that what is degree zero polynomial so if you see any constant so let's say if I say p of x is five times x to the power zero let's say we call it like that right so what is the degree here degree here is zero degree of px is zero okay now what happens if I have px is equal to zero so what is the degree of this degree of this now this is one exception polynomial exceptional polynomial please mind or please take note of it right so if px is equal to zero then degree is not defined degree of zero polynomial polynomial is not defined and why is that you can guess is not defined because zero can be written as anything so zero can be written as zero x to the power one million also right that doesn't mean this you know the degree becomes one million it would be anything zero into x to the power let's say minus not minus is not allowed so let's say 1000 so in both the cases the polynomial is same right but then we can't say that it has any arbitrary degree right so hence we say that degree of a zero polynomial is not defined zero polynomial means px equals to zero so for that particular polynomial no degree is defined okay in the next session we will be discussing about different types of polynomials how to be classified polynomials this is the degree