 Hello and welcome to the session. Let us discuss the following question. Verify that the numbers given along side of the cubic polynomial below are their zeros. Also verify the relationship between the zeros and the coefficient given to us is x cube minus 4x square plus 5x minus 2 and zeros are 2 comma 1 comma 1. So given to us is p of x is equal to x cube minus 4x square plus 5x minus 2 and zeros are 2 comma 1 comma 1. Now let alpha is equal to beta is equal to 1 and gamma is equal to 1. Now if 2 comma 1 comma 1 are the zeros of p of x then p of 2 is equal to 0, p of 1 is equal to 0 and p of 1 is equal to 0. Therefore, p of 2 is equal to, substituting x is equal to 2 in the given polynomial we get 2 cube minus 4 into 2 square plus 5 into 2 minus 2 equal to 0. Now solving this we get 8 minus 4 into 4 plus 10 minus 2 is equal to 0 which implies 8 minus 16 plus 10 minus 2 is equal to 0 which implies 0 is equal to 0. Hence p of 2 is equal to 0. Now similarly p of 1 is equal to substituting x is equal to 1 in the given polynomial we get 1 cube minus 4 into 1 square plus 5 into 1 minus 2 is equal to 0 which also we get 1 minus 4 into 1 plus 5 minus 2 is equal to 0 which implies 1 minus 4 plus 5 minus 2 is equal to 0 which implies 6 minus 6 is equal to 0 which implies 0 is equal to 0 hence p of 1 is equal to 0 hence 2 comma 1 comma 1 are the zeros of given polynomial p of x hence verified. Now let us do the second part of the question that is relationship between zeros and coefficients. Given to us p of x is equal to x cube minus 4 x square plus 5 x minus 2 comparing this with a x cube plus b x square plus c x plus d we get a is equal to 1 b is equal to minus 4 c is equal to 5 d is equal to minus 2. Now let alpha is equal to 2 beta is equal to 1 and gamma is equal to 1 we get alpha plus beta plus gamma is equal to 2 plus 1 plus 1 which is equal to 4 by 1 which is equal to minus b by a which is equal to minus coefficient of x square divided by coefficient of x cube. Similarly alpha beta plus beta gamma plus gamma alpha is equal to 2 into 1 plus 1 into 1 plus 1 into 2 which is equal to 2 plus 1 plus 2 which is equal to 5 by 1 which is equal to c by a which is equal to coefficient of x divided by coefficient of x cube. Similarly alpha beta gamma is equal to 2 into 1 into 1 which is equal to 2 by 1 which is equal to minus b by a which is equal to minus constant term divided by coefficient of x cube hence verified. I hope you have understood this problem. Bye and have a nice day.