 Hello and welcome to the session. In this session we discuss the following question which says, is the polynomial 6x to the power 4 plus 8x cube plus 17x square plus 21x plus 7 is divided by another polynomial 3x square plus 4x plus 1, the remainder comes out to be ax plus b, find a and b. Let's proceed with the solution now. Consider the polynomial px to be equal to 6 into x to the power 4 plus 8x cube plus 17x square plus 21x plus 7 and the other polynomial bqx be equal to 3x square plus 4x plus 1. We divide the polynomial px by the polynomial qx. Now when we divide 6x to the power 4 that is the first term of the dividend by 3x square which is the first term of the divisor. We get 2x square so we write 2x square in the quotient. Next we multiply this 2x square with every term of the divisor. So when 2x square is multiplied by 3x square we get 6x to the power 4. So we write 6x to the power 4 below 6x to the power 4 that is we write the like terms in a vertical column. Then 2x square is multiplied by 4x which gives us 8x cube. So we write 8x cube below 8x cube. Then 2x square multiplied by 1 gives us 2x square. It is written below 17x square. Now we subtract this that is 6x to the power 4 minus 6x to the power 4 is 0. Then 8x cube minus 8x cube is 0. Then 17x square minus 2x square is 15x square and we write the remaining terms of the dividend with 15x square. So we have 15x square plus 21x plus 7. Now next we divide 15x square by the first term of the divisor which is 3x square. This gives us 5 so we write plus 5 in the quotient. Now we multiply this 5 with every term of the divisor. 5 then multiplied by 3x square gives us 15x square and we write 15x square below 15x square. Then 5 is multiplied by 4x to give us 20x and we write 20x below 21x. 5 multiplied by 1 gives us 5 and we write plus 5 below 7. Now we subtract 15x square minus 15x square is 0. 21x minus 20x is x. Then 7 minus 5 is 2. So we have x plus 2 as the remainder. So we thus get the quotient as 2x square plus 5 and we have the remainder as x plus 2. But in the question we have that the remainder comes out to be ax plus b. So we compare ax plus b with x plus 2. That is we have ax plus b is equal to 1 into x plus 2. So when we compare these two terms we find that a is equal to 1 and b is equal to 2. So we have got the values for a and b. This is our final answer. This completes the session. Hope you have understood the solution of this question.