 Hey, hello friends welcome again to the session on polynomials and we are learning unique techniques of multiplication and division and in this case In this session. We are going to understand how to divide a polynomial by another polynomial In easier steps though it resembles very much the long division method But then we call it method of detached coefficients when you're dealing with theory of equations and higher polynomials little later Then these techniques become really really handy So what is this method of detached coefficients like in the previous session which we saw in wherein we Multiply two polynomials by just keeping the variables away and then simply dealing with the coefficients likewise Here also we will be dividing one polynomial by another one By simply, you know manipulating the coefficients and nothing else But before that as we did in the multiplications case We have to make sure that the polynomial polynomials are complete. That means all the powers of The variable is there in the polynomial. So let's first You know take this example Where in we have to divide x to power 8 plus x to power 7 plus 3 x to the power 4 minus 1 by x to the power 4 minus 3 x cube plus 4 x plus 1 clearly both of them are not complete So after completing if you see what will be the polynomials Like so this is x to power 8 x to power 7 0 x 6 and 0 times x 5 and 3 times x to the power 4 Then we have a plus 0 times x cubed plus 0 times x squared Plus 0 times x and then minus 1. This is a very long polynomial. No problem and You know the divisor is so this was dividend and This is divisor divisor is x to power 4 minus 3 x cubed plus 0 x squared plus 4 x Plus 1 isn't it? Looks very simple now how to go about it. So obviously you can't be writing all those Powers of x again and again. So like you used to do it. So let's first write the divisor on the left-hand side So and only the coefficients will attach the coefficients That means this becomes one and just keep a track of all the devices one minus three zero four and one These are the coefficients of the divisor right and let's write the coefficients of the dividend one one zero zero Three zero zero zero minus one Correct. So let it be like that. I Hope this is understood to you. So we have this detached the coefficients and wrote them in a line now what now you follow the usual division method So hence, let's say first this is the target one and we have to make sure what should I multiply this one here So that I get one in this case. So simply write one so one This one what all what all things are getting multiplied. So this one is getting multiplied by this one To fetch you this one. So just like another, you know, so it's like I'm doing the same long division method But I have just kept the variables away. That's it. Okay, so Simply if I write one then my problem is this minus three zero four one then subtract Usually the way we follow. So four zero minus four and two Okay, and then take this zero down this one here comes down here. Okay, then next one. Definitely, this will be plus four Don't write, you know, plus in, you know, like that it will confuse you so like for only so four times one is Four then minus three minus twelve zero sixteen and four I hope this is becoming on or this is not, you know, something which you can't understand again subtract So this is zero. This is 12. This is minus four. This is minus 14 and this is minus four And now this zero comes here. So Again, what should be the next one? Okay, so clearly zero Oh, sorry, no zero not zero. Sorry. This one is the first one now. So what should I what should I write here? 12 clearly so 12 will give you 12 minus 12 at 12 times minus three is minus 36 Okay, so I hope you're understanding. So this 12 is getting multiplied by one first then minus three then zero then four Then one like that. So 12 times minus three is minus 36 12 times zero is zero 12 times four Is 48 and this is 12 subtract again First one gets cancelled. This becomes 32. This becomes minus 14 This becomes minus 52 And this becomes minus 12 and then bring down this zero Zero Now what very clearly this should be 32. So when 32 times one is 32 Then 32 times minus three is minus 96 Correct, then it is zero then it is 32 times four is 128 and 32 times one is 32 Okay, now subtract again. So what will happen? This will go And this will be plus 82. Yep. This will be minus 52 This will be minus 14 zero and this is minus 32 and then bring down this minus Okay, so minus one. Okay. Now what 82 Multiply by 82. So you get 82 82 times three is 246 so minus 246 82 times zero is zero 82 times four is Um, 328. Yes, 328 And 82 times one is 82. So now What is the This thing leftover thing. This is the one this 246 minus 52 is one sorry, plus one nine four And this one is minus one four zero This one is minus three six zero And this one is minus 83 Okay, now you no more can divide. So now you would have guessed by you know, what would be the quotient and what would be the remainder So my dear friends, what would be the quotient and what would be the remainder? Okay, now quotient is simply So you have to start with x to the power of zero Then plus x to the power one Then this is x squared x cubed and x to the power four Right simply now attach. So while we detach the coefficients now we attach here also It will be 83 and then minus 360 x minus 140 x square and x cube Okay, so you have to write quotient is x to the power of four plus four x cubed Plus 12 x squared plus 32 x plus 82 This one is your quotient and what is the remainder? Reminder is one nine four x cubed Minus one four zero x squared Minus three six zero x Minus 83 Okay, so this is quotient. This is a remainder. So though so what is the learning in this process? It's not different from long division. But the only thing is You need not keep a Track of all the variables you can just detach the coefficient and go for The division which you have been doing so far, right? So it will save some of your Efforts in the next session. We'll learn synthetic synthetic division of polynomial by linear polynomials