 Hello and welcome to the session. In this session we discussed the following question which says if px is equal to x cube plus ax square plus bx plus 6 leaves the remainder 3 when divided by x minus 3 and x minus 2 is the factor of px find the values of a and b. Before moving on to the solution let's recall the remainder theorem first. According to this theorem we have let px be any polynomial of degree n greater than or equal to 1 and a be any real number and if the polynomial px is divided by the linear polynomial x minus a then the remainder is pa. Next we have the factor theorem according to this we have that is px is a polynomial of degree n greater than equal to 1 and a be any real number then we have x minus a is a factor of px if pa is equal to 0 and pa equal to 0 if x minus a is a factor of this is the key idea to be used for this question. Now let's move on to the solution we are given the polynomial px is equal to x cube plus ax square plus bx plus 6 it's given that when the polynomial px is divided by x minus 3 then the remainder is equal to p3 according to the remainder theorem we have that is polynomial px is divided by x minus a then it's remainder is pa that is when we apply the remainder theorem in this case we say that when polynomial px is divided by x minus 3 then remainder is equal to p3. Now we have p3 is equal to 3 putting x equal to 3 in this polynomial px we get p3 that is 3 cube plus a into 3 square plus b into 3 plus 6 is equal to 3 this gives us 27 plus 9 a plus 3 b plus 6 is equal to 3 which further gives us 9 a plus 3 b is equal to minus 30 or we can say 3 a plus b is equal to minus 10 let this be equation 1. Now the next condition given to us in the question is that x minus 2 is a factor of px now recall the factor theorem in this we have one condition that if x minus a is a factor of px then we have pa is equal to 0. Now where x minus 2 is a factor of px so this would mean p2 is equal to 0 putting x equal to 2 in this we get p2 so we have this would mean 2 cube plus a into 2 square plus b into 2 plus 6 is equal to 0 that is we have 8 plus 4 a plus 2 b plus 6 is equal to 0 this further gives us 4 a plus 2 b is equal to minus 14 or we can say 2 a plus b is equal to minus 7 let this be equation 2. Now we will solve the equation 1 and 2 to find the value for a and b now we have 3 a plus b equal to minus 10 and 2 a plus b equal to minus 7 now we subtract the second equation from the first equation so we get a is equal to minus 3. Now substituting a equal to minus 3 in equation 1 we get 3 into minus 3 plus b equal to minus 10 that is we get minus 9 plus b equal to minus 10 which further gives us b equal to minus 10 plus 9 so we get b equal to minus 1 so now we have got a equal to minus 3 and b equal to minus 1 so this is our final answer with this we complete the session hope you have enjoyed the session.