 Hello and welcome to the session. In this session we discussed the following question which says the polynomial fx equal to x to the power 4 minus 2xq plus 3x square minus ax plus b when divided by x minus 1 and x plus 1 leaves the remainder 4 and 2 respectively find the values of a and b. Before moving on to the solution let's recall the remainder theorem according to this we have let px be any polynomial of degree n greater than equal to 1 and let a be any real number. Now if the polynomial px is divided by some linear polynomial x minus a then the remainder is p a. This is the key idea to be used for this question. Let's proceed with the solution now we are given the polynomial fx is equal to x to the power 4 minus 2xq plus 3x square minus ax plus b. Now it's given when fx is divided by x minus 1 the remainder would be equal to f of 1 using the remainder theorem that is when px divided by x minus a its remainder is p a. So in this case when this fx is divided by x minus 1 its remainder would be f1. Now in the question it's given when fx is divided by x minus 1 its remainder is 4 and here we have remainder is f1 so we say f1 is equal to 4. Now how do we get f1? f1 is given by substituting x equal to 1 in this fx. So we get 1 to the power 4 minus 2 into 1q plus 3 into 1 square minus a into 1 plus b equal to 4 that is we have 1 minus 2 plus 3 minus a plus b is equal to 4 that is we get minus a plus b is equal to 2. Let this be equation 1. Now next we have when fx is divided by x plus 1 then the remainder would be equal to f of minus 1 using the remainder theorem and in the question it's given that fx when divided by x plus 1 gives us the remainder 2. So this would mean that f of minus 1 is equal to 2. Now we substitute x equal to minus 1 in this fx to get f of minus 1. So we get minus 1 to the power 4 minus 2 into minus 1 to the power 3 plus 3 into minus 1 to the power 2 minus a into minus 1 plus b is equal to 2 that is we get 1 plus 2 plus 3 plus a plus b is equal to 2 that is we have a plus b is equal to minus 4. Let this be equation 2. Now we shall solve equations 1 and 2 to get the values for a and b. So we add equations 1 and 2. So on adding 1 and 2 we get 2b is equal to minus 2 that is b is equal to minus 1. Now substituting b equal to minus 1 in equation 1 we get minus a plus minus 1 is equal to 2 that is we get minus a is equal to 2 plus 1 that is a is equal to 3 or we say a is equal to minus 3. So final answer is a equal to minus 3 and b equal to minus 1. This completes this session. Hope you have enjoyed this session.