 Hello and welcome to the session. Today I am here to help you with polynomials, so let it is called the following question. If the polynomial x4 minus 6xq plus 16x square minus 25x plus 10 is divided by another polynomial x square minus 2x plus k, the remainder comes out to be x plus a, find k and a. So let us first write the key idea that we will be using another problem. According to division algorithm we have p of x equal to g of x into q of x plus r of x, that is, dividend is equal to divisor into quotient plus remainder. Now let us write our solution. Given to us this p of x is equal to x4 minus 6xq plus 16x square minus 25x plus 10 and g of x is equal to x square minus 2x plus k and r of x is equal to x plus a. We have to find k and a. By key idea we have p of x is equal to g of x into q of x plus r of x or p of x taking this rxx to left hand side so we get minus r of x is equal to g of x into q of x. Now substituting the values we get x to the power 4 minus 6xq plus 16x square minus 25x plus 10 minus x plus a is equal to x square minus 2x plus k into q of x or x4 minus 6xq plus 16x square minus 25x plus 10. Opening this bracket we get minus x minus a is equal to x square minus 2x plus k into q of x. Now solving it further we get x4 minus 6xq plus 16x square minus 25 minus x is 26x plus 10 minus a is exactly divisible by x square minus 2x plus k. Now we divide x4 minus 6xq plus 16x square minus 26x plus 10 minus a by x square minus 2x plus k by using long division method. Therefore our evident will be x4 minus 6xq plus 16x square minus 26x plus 10 our divisor will be x square minus 2x plus k. Therefore now dividing it we get x4 divided by x square gives you x square minus 2xq plus kx. Now solving this 26x plus again minus 4xq divided by x square gives minus 4x minus 2x into minus 4x plus 8x square into k gives. Now again subtracting it don't forget to change the signs into 26 minus 4k divided by x square gives into k gives plus k plus k square and our quotient as x square equal to 0 equal to 0. Solving this we get 2k is equal to 10 which implies k is equal to 5 k is equal to 5. Now solving this we get 10 minus a substituting the value of k is equal to 5 from here we get minus 8 into 5 plus 5 square equal to 0 which implies 10 minus a minus 40 plus 25 equal to 0 which implies minus a minus 5 is equal to 0 which implies a is equal to minus 5 therefore a is equal to minus 5. Hence required equal to 5 and a is equal to minus 5. Hope you understood this problem. Bye and have a nice day.