 Hi and welcome to the session. My name is Shashi and I am going to help you to solve the following question. Question is, give examples of polynomials px, gx, qx and rx which satisfy the division algorithm and degree of px is equal to degree of qx. First of all we should know that dividend is equal to divisor multiplied by quotient plus remainder that is px is equal to gx multiplied by qx plus rx. Where px is the dividend, gx is the divisor, qx is the quotient and rx is the remainder. Expression is known as division algorithm. This is the key idea to solve this question. Let us start with the solution now. We know we have to obtain a polynomial such that degree of px is equal to degree of qx. We know in division algorithm px is equal to dividend and qx is equal to quotient. Now let px is equal to 3x cube minus 6x square plus 12x plus 6. Here the degree of px is equal to 3, right? Now let gx is equal to 3. This implies degree of gx is equal to 0. So to obtain the degree of px and the degree of qx equal we had taken the degree of px equal to 3 and the degree of gx equal to 0. Let us start the division now. We can see first term of the dividend is 3x cube so we will multiply 3 with x cube to obtain the desired term. Now 3x cube and 3x cube get cancelled and bringing down the rest of the terms of the dividend we get minus 6x square plus 12x plus 6. Now our first term is minus 6x square so we will multiply 3 with minus 2x square to get minus 6x square. Now the light terms will get cancelled and we will get 12x plus 6 after rewriting the rest of the terms of the dividend. Now to get 12x we multiply 3 with 4x. Now again subtracting the light terms we get 0 plus 6. 6 is the left term in the dividend. Now to get 6 we will multiply 3 with plus 2. Now multiplying 3 with plus 2 we get 6 and subtracting the light terms again we get the remainder as 0. Therefore our qx is equal to x cube minus 2x square plus 4x plus 2, px is equal to 3x cube minus 6x square plus 12x plus 6, gx is equal to 3 and rx is equal to 0. Let us now check if the values of the px, gx, qx and rx satisfy the division algorithm. Taking rhs of the division algorithm we get qx multiplied by gx plus rx. Now substituting the values of qx, gx and rx we get x cube minus 2x square plus 4x plus 2 multiplied by 3 plus 0 which is equal to 3x cube minus 6x square plus 12x plus 6. This is equal to lhs equal to px. Also the degree of px and the degree of qx is equal to 3. So this implies px, qx, gx and rx satisfy the division algorithm and also the degree of px is equal to degree of qx. This completes our session. Hope you enjoyed this session. Bye bye.