 Hey hello friends welcome again to another session on polynomials and in the last few sessions we have been seeing how to find out alternate methods of multiplication and division. Continuing with that trend today in this session we are going to discuss synthetic division of polynomials by linear polynomials right. So this is a specific case where we have been given a very big polynomial large expression and we have to divide it by x plus two and find quotient and remainder. So please remember this particular division method works only with when the divisor is linear okay. So you can see this is linear y because the power is one here. So this is a linear divisor and dividend could be anything and we will now learn how to quickly do it within no time right. Okay so as we have been seeing in all these cases you have to first ensure that the polynomial is complete right. That means all the powers of polynomials are there and then you have to detach the coefficient. Now if you look closely here there is no x cube so hence the complete polynomial would be 3 to the power 3xx minus 7x to power 5 plus 5x to the power 4 minus 0x cubed minus x squared minus 6x or this could be plus 0x cube as well minus 6x minus 8 okay. So this is what is the complete polynomial. So once that is done you simply write all the detached coefficients. So in this case it is 3 then space out minus 7 then 5 then 0 then minus 1 minus 6 and finally minus 8 okay. Then what you need to do is now look at the divisor so x plus 2 express this as x minus minus 2. So it has to be expressed in x minus alpha form x minus alpha form right. So it is x plus 2 so I can say x minus minus 2 right. If it was x minus 1 so then you don't need to do anything it is x minus 1 directly right. If the divisor was x minus 1 since it is x plus 2 so I have to write x minus minus 2 and then take this minus 2 here and draw a line okay. This is how you have to begin. Now how to find out or how to go about this division. What you need to do is you simply write this 3 down here first no nothing right 3 down. Now what you take and divide this oh sorry multiply this minus 2 times 3 what is it you will get minus 6 write that minus 6 here okay and then add the column what will you get you will get minus 13 isn't it. Then again what take minus 2 and multiply minus 13 with it what is it 26 where will you write here okay add the column you will get 31 isn't it. Then what take minus 2 multiply this 31 what is it minus 62 where will you write here minus 62 okay then what add the column add so minus 62 then what again repeat the process go to this minus 2 times minus 62 is 1 2 4 where where will you write it here 1 2 4 add 2 add this column you will get minus oh sorry 1 2 3 isn't it then what again minus 2 and multiplied by this 123 value is minus 246 write this here okay now add this column what will you get minus 25 correct. Then one more time take this and then multiply what will you get 504 write 504 and then add the column what will you get you will get 496 correct 496 so hence the last term is the remainder this is the remainder and now what is the quotient quotient is simply now attach the coefficients back so minus 252 will be the constant term 123x minus 62x squared then 31x cubed minus 13 x to the power 4 and 3 x to the power 5 this is your quotient remainder is 496 got it this is called synthetic division so you can do it mentally also if you have a lot of practice done right so this is called synthetic division I will take one more example to show you this how to divide let's say you have 5x to the power 5 minus 7x cube okay then you have 6x squared let's say then you have minus 2x and let's say then you have 4 and you have to divide this by by x minus 1 simpler one right so it's already in x minus if alpha form so what do you need to do I will quickly write 1 because that's my you know factor here to be used and then complete polynomial all coefficients of 5 then x4 coefficient is missing so write 0 then minus 7 then 6 then minus 2 and then 4 and start how to start you simply write this 5 here then 1 times 5 is 5 write it here okay and add you'll get 5 then again 1 times this 5 how much 5 write it here 5 so what is if you add you'll get minus 2 is it then what then 1 times minus 2 is how much minus 2 add 4 then I'm not drawing any arrows further so 1 into 4 is 4 so 2 and then 1 into 2 is 2 so hence 6 so you got the remainder remainder is 6 and your quotient is 2 4x then minus 2x squared just attach the coefficient back x cubed and then 5x4 done so this is the quotient 5x to the power 4 plus 5x cube minus 2x squared plus 4x plus 2 and the remainder is 6 so can you see how quickly you could do this division this process is called synthetic division