 hello friends so welcome to this new session on polynomials and today we are going to discuss division algorithm for polynomials so if you remember you've already studied about division algorithm in arithmetic or number system where we studied about something called Euclid's Euclid's division lemma isn't it if you remember Euclid's division lemma we studied in the real numbers topic or number theory basically we studied this in number theories now Euclid's division lemma was nothing but if a and b are two a and b are two positive integers so usually we talk about positive integers positive integers integers then we we know that a is equal to bq plus r right this was Euclid's division lemma where where r was greater than equal to 0 but less than equal to b right this is what was Euclid's division lemma which in other words you have been studying this is like dividend dividend in a process of division is equal to divisor divisor times divisor time quotient plus remainder isn't it now the same laws also is applicable in case of polynomials as well so let us try with an example and then we'll come to the formal definition of division algorithm of polynomials okay so let me take an example let us say I want to divide a polynomial fx which is given by x cube minus 6x squared plus 11x minus 6 okay so some basic information about this this is certainly a polynomial and what is the degree of fx if you see degree of fx is nothing but 3 right highest power of x is 3 so degree of fx is 3 now let us say I have to divide this fx by another function gx or another polynomial gx which is x plus 2 okay what is the degree of gx is equal to 1 right now let us divide it now let us do the long division method so how do you divide it you remember so this is how you have to write that dividend here x cube minus 6x square plus 11x minus 6 so hence first term will be x squared why because x into x squared will be x cube so x cube and then plus 2x square is it 2x square now you subtract you will get this is would be minus sign right so it is minus 8x square plus 11x minus 6 what next then minus 8x is it it so minus 8x the moment you do you will get minus 8x square here then minus 16x is it it and then again you have to perform the subtraction so it will be 27x minus 6 right 27x minus 6 then what then you have simply do 27 so you'll see it is 27x and then plus 54 right and then you subtract again so what do you get you get this as 0 and 6 minus 6 minus 54 is minus 60 isn't it so you see you get remainder as minus 60 and the quotient is x square minus 8x plus 27 right this is the quotient so I can write this as quotient and this has remainder now you if you if you want to verify you can see that you can very well see that x cube minus 6x square plus 11x plus sorry minus 6 minus 6 is equal to this is the dividend isn't it this is the dividend and what is the divisor x plus 2 was the divisor times quotient x square minus 8x plus 27 plus the remainder that is minus 60 and if you if you if you see you can you can try and verify you will if you multiply these two and then add minus 60 to it you will get this term right so hence here it is dividend is equal to divisor into quotient plus remainder now let us you know draw some conclusions if you say what is the degree of quotient right degree of quotient if you say degree of quotient is less than that of the dividend okay it might be true it might not be true as well because if there's a constant term as a quotient then the degree will not be so but in any case degree of the quotient will be either equal to or less than the dividend similarly divisor the degree of divisor is x plus 2 which is less than that of dividend and the most important point is the degree of the remainder now degree of remainder of remainder is always less than degree of divisor and why is it so because if it was not so then you could have gone for one more step of division what do i mean let us say instead of minus 60 here if you would have got minus 60x let us say you got minus 60x then what you would have again gone for another set another round of division isn't it you would have written minus 60 here and then it would be minus 60x and then minus 120 so again you would have performed one more division and you would have gone got a remainder which whose degree is less than that of divisor okay so please understand that degree of remainder will always be less than degree of divisor and this is nothing but in a nutshell we discussed something called division algorithm for polynomial now let us see definition of division algorithm and it says if fx and gx are two polynomials are two polynomials two polynomials with gx not equal to zero now gx must not be zero why because if gx is equal to zero then division is not defined then then we can always find always find qx and rx such that such that and mind you qx and rx are polynomials okay such that fx is equal to gx times qx plus rx and and and degree of rx is less than degree of degree of sorry gx this is what is the formal definition of degree of gx okay this is what is the definition of division algorithm for polynomial so fx is a polynomial you divide this polynomial by gx you will get a quotient and you will get a remainder such that the degree of the remainder is less than that of degree of the divisor okay let's take an example and understand we have already seen one example let us take another example so let us take another example to see this now the question is let us say we have fx is equal to x to the power four minus three x squared plus four x plus five now what is degree of fx degree of fx here is four and the divisor gx is let us say x square minus x plus one okay and a degree of gx is equal to two correct so hence I can always find as per the division algorithm fx must be equal to gx times qx plus rx okay where degree of rx will be less than degree of gx so what is the degree of gx two so hence degree of rx will be less than two that means either it is a linear factor linear polynomial right hence only its degree is one or it is a constant so degree is zero correct is it so let us you know try and divide and see whether we are getting that is that result so x square let us perform the long division method x square minus x plus one x four minus three x square plus four x plus five okay let's divide so first term clearly will be x square so hence it is x to the power four minus x cube there will be one minus x cube right and plus x square plus x square isn't it so hence if you subtract this will get cancelled and you will get x cube and this is minus four x squared plus four x plus five then next term will be certainly x isn't it so if you multiply by x you will get x square x cube then minus x square is it and then plus x subtract again you will get minus three x square and then plus three x plus five isn't it then what now it will be minus three so minus three x squared then what plus three x and minus three is it so if you do the calculation you will get only eight as the remainder isn't it so hence we see x to the power four minus three x square plus four x plus five can be written as x square minus x plus one times x square plus x minus three plus eight right so this is nothing but this is my fx this is gx divisor this is qx the quotient and this is the remainder rx and clearly degree of rx is two right which is less than degree of gx correct this is what was division algorithm now in the next session we'll learn how to find out the quotient and remainder without doing actual division okay