 Hello and welcome to this session. Let us discuss the following problem today. Verified as the numbers given alongside of the cubic polynomials below are their 0s. Also verify the relationship between the 0s and the coefficients. We have 2xq plus x2 minus 5x plus 2 and 0s as 1 by 2 comma 1 comma minus 2. Now let us understand the key idea. A real number alpha is a 0 for polynomial f of x if f of alpha is equal to 0. A polynomial of degree n can have at most n real 0s. Thus a cubic polynomial can have at most 3 real 0s. If alpha, beta, gamma are the 0s of the cubic polynomial A x cube plus B x square plus C x plus T, then alpha plus beta plus gamma is equal to minus B by A. And alpha beta plus beta gamma plus gamma alpha is equal to C by A. And alpha beta gamma is equal to minus C by A. Now let us proceed with our solution. Given to us this p of x is equal to 2x cube plus x square minus 5x plus 2 and given 0s are half comma 1 comma minus 2. Now if half comma 1 comma minus 2 are the 0s of p of x then p of half equal to 0, p of 1 is equal to 0 and p of minus 2 is equal to 0. Therefore p of half is equal to substituting x is equal to half in the given polynomial p of x. We get 2 into half the whole cube plus half square minus 5 into half plus 2. Now is equal to 0. Now solving this further we get 2 into 1 by 8 plus 1 by 4 minus 5 by 2 plus 2 is equal to 0. This gets cut off. So we get now 1 by 4 plus 1 by 4 minus 5 by 2 plus 2 is equal to 0. Now solving it further taking else name we get 1 plus 1 minus 10 plus 8 is equal to 0 which implies 0 is equal to 0. Therefore p of half is equal to 0. Similarly p of 1 substituting x is equal to 1 in our given polynomial we get 2 into 1 cube plus 1 square minus 5 into 1 plus 2 is equal to 0. Solving this further we get 2 into 1 plus 1 minus 5 plus 2 is equal to 0 which implies 2 plus 1 minus 5 plus 2 is equal to 0 which implies 0 is equal to 0. Hence p of 1 is equal to 0. Now similarly p of minus 2 substituting x is equal to minus 2 in given polynomial p of x we get 2 into minus 2 to the power whole cube plus minus 2 square minus 5 into minus 2 plus 2 is equal to 0. Solving it further we get 2 into minus 8 plus 4 plus 10 plus 2 is equal to 0 which implies minus 16 plus 4 plus 10 plus 2 is equal to 0 which implies minus 16 plus 16 is equal to 0 which implies 0 is equal to 0. Hence p of minus 2 is equal to 0. Therefore comma 1 comma minus 2 are the required given polynomial p of x hence verified. So the 22nd part of the question that is verifying relationship between coefficients. So given to us p of x is equal to 2x cube plus x square minus 5x plus 2. Now comparing this p of x with ax cube plus bx square plus cx plus d we get a is equal to 2, b is equal to 1, c is equal to minus 5 and d is equal to 2. Now let alpha is equal to half, beta is equal to 1 and gamma is equal to minus 2. So therefore alpha plus beta plus gamma is equal to half plus 1 minus 2 which is equal to minus 1 by 2 which is equal to minus b by a which can be written as minus coefficient of x square divided by coefficient of x cube. Similarly alpha beta plus beta gamma plus gamma alpha is equal to half into 1 plus 1 into minus 2 plus minus 2 into half this gets cut off. So we get half minus 2 minus 1 which is equal to 1 minus 4 minus 2 divided by 2 which is equal to minus 5 by 2 which is equal to c by a which can be written as coefficient of x divided by coefficient of x cube. Now similarly alpha beta gamma is equal to 1 into minus 2 which is equal to minus 1 which is equal to minus 2 by 2 which is equal to minus d by a which can be written as constant term divided by coefficient of x cube hence verified. I hope you understood the problem. Bye and have a nice day.