 Hello and welcome to this session. In this session we discuss the following question which says, divide 3x cube minus 8x square plus 9x minus 3 by x square minus 2x plus 2 and verify the division algorithm. So as you can see in the question we are given two polynomials, we need to divide the two polynomials and we also need to verify the division algorithm. So first let's state the division algorithm. According to this we have if px and gx are any two polynomials with gx not equal to 0 then we can find polynomials qx such that x is equal to gx into qx plus rx where we have rx is equal to 0 or degree of rx is less than degree of gx. So we use this division algorithm as the key idea for the solution of this question. Now we proceed with the solution. We suppose let the polynomial px be equal to 3x cube minus 8x square plus 9x minus 3 and the polynomial gx be equal to x square minus 2x plus 2. First of all we need to see that if the given polynomials are in standard form of decreasing order of degrees as you can see that the polynomial px is in the standard form that is in decreasing order of the degrees and the polynomial gx is also in the standard form in decreasing order of the degrees. So we will now divide both these polynomials that is we will divide the polynomial px by polynomial gx. We write the polynomial gx outside that is x square minus 2x plus 2 and the polynomial px inside that is 3x cube minus 8x square plus 9x minus 3. Now first of all consider the first term of the polynomial px that is 3x cube divide this by the first term of the polynomial gx that is x square. So when you divide 3x cube by x square you get 3x. So we write this 3x in the quotient and we multiply this 3x with each term of this polynomial gx. So 3x when multiplied by x square gives us 3x cube. So we write it below 3x cube since they are the light terms. Now 3x multiplied by minus 2x gives us minus 6x square. We write it below minus 8x square since they are the light terms. Then 3x multiplied by 2 gives us plus 6x and we write it below 9x since they are the light terms. Now we will subtract them. So 3x cube minus 3x cube cancels. This gives us 0. Then minus 8x square plus 6x square gives us minus 2x square then plus 9x minus 6x gives us plus 3x. Now we write this minus 3 along with this minus 2x square plus 3x. So in the same way we will divide this new polynomial that is minus 2x square plus 3x minus 3 by x square minus 2x plus 2. So we will consider the first term of this polynomial that is minus 2x square and we divide this by x square. So minus 2x square when divided by x square gives us minus 2. So we write minus 2 in the quotient. Now we multiply this minus 2 with each term of this polynomial gx. So minus 2 multiplied by x square gives us minus 2x square and we write it below minus 2x square. Then minus 2 multiplied by minus 2x gives us plus 4x and we write it below 3x since they are the light terms. Then minus 2 multiplied by 2 gives us minus 4 and we write it below minus 3. Now we will subtract them. So this would give us 0 minus x plus 1. So we now get the quotient that is qx is equal to 3x minus 2 and the remainder rx is equal to minus x plus 1. Now as you can see that degree of rx that is minus x plus 1 is 1 which is less than the degree of the polynomial gx which is x square minus 2x plus 2 and its degree is 2. So we had this condition in the divisional algorithm that we have stated in the key idea that degree of rx is less than gx or rx should be equal to 0. But since we have got that rx is not equal to 0 so the other condition is satisfied that is degree of rx is less than degree of gx. Now we have got all the polynomials polynomial px polynomial gx then qx and rx. Now we need to show the verification of division algorithm that is we need to show that the polynomial px is equal to polynomial gx into polynomial qx plus polynomial rx. Now first we consider polynomial gx multiplied by polynomial qx plus polynomial rx this would be equal to x square minus 2x plus 2 multiplied by polynomial qx which is 3x minus 2 plus polynomial rx which is minus x plus 1. So this would be equal to 3x cube minus 2x square minus 6x square plus 4x plus 6x minus 4 plus minus x plus 1 this is equal to 3x cube minus 8x square plus 10x minus 4 minus x plus 1 that is we have this would be equal to 3x cube minus 8x square plus 9x minus 3 and as you know that this is equal to the polynomial px that is we have got polynomial px is equal to gx into qx plus rx so hence division algorithm is verified. So this completes the session hope you have understood the solution for this question.