 Hi and welcome to the session. Today we will discuss the following question. The question says find a cubic polynomial with the sum, sum of the product of its zeros taken 2 at a time and the product of its zeros as 2 minus 7 minus 14 respectively. Before proceeding for the solution, recall that if alpha, beta, gamma are the zeros of the cubic polynomial AXQ plus BX square plus CX plus D equal to 0, then alpha plus beta plus gamma is equal to minus B upon A alpha beta plus beta gamma plus gamma alpha is equal to C upon A and alpha beta gamma is equal to minus D upon A. This is the key idea for this question. Now let's see its solution. In this question we need to find a cubic polynomial. So we will assume the cubic polynomial to be AXQ plus BX square plus CX plus D. So let the cubic polynomial be P of X equal to AXQ plus BX square plus CX plus D of beta gamma are the zeros of the cubic polynomial P of X. Now as we are given in the question, the sum of the zeros is 2. So this means we have from the question alpha plus beta plus gamma is equal to 2. Also it's given that the sum of the product of its zeros taken 2 at a time is minus 7. So this means alpha beta plus beta gamma plus gamma alpha is equal to minus 7. The product of its zeros is minus 14. So this gives us alpha into beta into gamma is equal to minus 14. Also from the key idea we have alpha plus beta plus gamma is equal to minus B upon A. So let's write it over here alpha plus beta plus gamma is equal to minus B upon A. Now alpha beta plus beta gamma plus gamma alpha is equal to C upon A. So we have alpha beta plus beta gamma plus gamma alpha is equal to C upon A and alpha beta gamma is equal to minus D upon A. Now we will compare these two columns. So from here we have alpha plus beta plus gamma equal to 2. Also alpha plus beta plus gamma is equal to minus B upon A. So comparing these two we get 2 is equal to minus B upon A or we can also write it as minus B upon A equal to 2 upon 1 which is equal to minus of minus 2 upon 1. So this gives us B is equal to minus 2 and A is equal to 1. Now we have alpha beta plus beta gamma plus gamma alpha is equal to minus 7. Also this is equal to C upon A. So comparing these we get minus 7 is equal to C upon A or we can write it as C upon A is equal to minus 7 upon 1. So this implies that C is equal to minus 7, A is equal to 1. Now from this we get the alpha beta gamma is equal to minus 14 and minus D upon A. So these two gives us minus 14 is equal to minus D upon A or we can write it as minus D upon A is equal to minus 14 upon 1. This implies that D is equal to 14 and A is equal to 1 therefore we have A equal to 1, B equal to minus 2, C equal to minus 7 and D equal to 14. Now our cubic polynomial B of x is equal to A x cube plus B x square plus C x plus D. So substituting the values of A, B, C and D we will get 1 x cube plus B that is minus 2 x square plus C x that is minus 7 into x plus D that is 14. So this is equal to x cube minus 2 x square minus 7 x plus 14. Therefore the required cubic polynomial is x cube minus 2 x square minus 7 x plus 14. With this we finish this session. Hope you must have understood the question. Goodbye, take care and have a nice day.