 Hello and welcome to another problem-solving session on factorization of polynomials right the question is Find the remainder when py is equal to y cube minus 6 y square plus 2 y minus 4 is Divided by Q y that is 3 y minus 1 so you can see we have a polynomial p y It is polynomial in variable y okay, so it can be variable any variable polynomial So here the variable is why it could be x z d s whatever right so this is a given polynomial and my divider Right divisor is 3 y Minus 1 right so by remainder theorem By remainder theorem What did we learn in the previous sessions By remainder theorem if FX is Divided by AX plus BA linear divisor Right, please understand. This is a only for is only for linear divisor Okay linear means the power on X must be only one or the variable must be one remainder is equal to F minus B by A Correct the remainder is F of minus B by a what is B and a we can find out So clearly in this case a is 3 because our divisor was what was the divisor? divisor is 3 y minus 1 Variable is why so a is 3 B is minus 1 so minus B by a is equal to minus of minus 1 by 3 that is 1 by 3 Correct so 1 by 3 so hence we have to simply now find out F of that is minus B by a and in this case F is capital P. So P of 1 by 3 right P of 1 by 3 will be the remainder Right, I hope this is understood remainder is nothing but F of F of our value of the polynomial at X equals to minus B by a in this case variable is why so y is equal to minus B by a minus B by a is 1 by 3 Right, so P of 1 by 3 if we calculate we get the remainder Okay, so what is P of 1 by 3? Let's evaluate so 1 by 3 whole cube So this is the you know, so you have to just deploy 1 by 3 in place of y minus 6 1 by 3 whole squared plus 2 times 1 by 3 minus 4 This is a calculation. That's it. This is the remainder also, right? So this is 1 upon 27 minus 6 upon 9 right minus minus 2 by 3 minus 4 Right, so what is the common denominator? 27 LCM is 27. So here it is 1 minus you can This is nothing but 9 into 3 6 18 minus 18 again isn't it minus 4 into 20 7 will be 108 Okay, I hope this is correct calculation. So 1 Just a minute minus this is plus sorry, this is plus Right, so 1 by 27 minus 6 by 9 plus 2 by 3 minus 4. Yes, so this minus 18 plus 18 goes This is nothing but minus 1 not 7 upon 27 Okay, so if you divide Divide the Px. So if you divide Px By this 3x minus 1 you will get a quotient. Let's say Gx Or this is why sorry, this is why so P of y is 3 y minus 1 into some g y minus 107 by 27 Right, so this is the remainder Okay, P y was my dividend 3 y minus 1 was divisor This one is quotient What is that quotient? We don't know but we are not interested either and this one is remainder isn't it this is the Reminder so we caught we could find out the remainder without actual long division method This is the application of remainder theorem