 Hi and welcome to the session, I am Asha and I am going to help you solve following question which says verify whether the following are zeros of the polynomials indicated I mean spin. So first let us learn what is zero of the polynomial. Suppose we have a polynomial px then x is equal to c is called the zero of px if c is equal to zero. So this is the key idea we are going to use in this problem to find its solution. So all that you have to do to verify whether the certain given value of the variable is a zero of the polynomial or not just replace the variable in the given polynomial by the given value of the variable. Then if say value c gives pc is equal to zero where px is any polynomial then c is called the zero of the polynomial. Now let us see the first one which is px is equal to 3x plus 1 and we have to check that if x is equal to minus 1 upon 3 is a zero of the polynomial px or not. So on replacing x by minus 1 upon 3 in the polynomial px we have 3 into minus 1 upon 3 plus 1 and 3 cancels with 3 we have minus 1 plus 1 which is equal to zero. Thus we have p at minus 1 upon 3 is equal to zero. So we say that x is equal to minus 1 upon 3 is a zero of the given polynomial which is px is equal to 3x plus 1. So this completes the first part. Now proceeding on to the next one which is px is equal to 5x minus pi where x is equal to 4 upon 5. Now to check whether x is equal to 4 upon 5 is a zero of px or not let us substitute x by 4 by 5 and 5x minus 3. Now x is 4 upon 5 minus pi which is equal to 22 upon 7. 5 cancels out with 5 and we have 4 minus 22 upon 7. Now 4 can be written as 4 upon 1 and upon taking LCM of 7 and 1 we have 7 and in the numerator on dividing 7 by 1 we get 7. So multiplying the numerator 4 with 7 we have 28 minus 22 which is equal to 6 upon 7. Thus p at 4 upon 5 is equal to 6 upon 7 which is not equal to zero. Hence it is not a zero of the given polynomial px. As an answer is no, x is equal to 4 upon 5 is not a zero of the polynomial px which is equal to 5x minus pi. So this completes the second part. And now proceeding on to the third one where px is equal to x square minus 1 and we have to check whether x is equal to 1 and x is equal to minus 1 are the zeros of px or not. So first let us substitute x is 1 then we have 1 square minus 1 which is equal to 1 minus 1 which is further equal to 0. Thus we have p at 1 is equal to 0 and now let us find p at minus 1. So we have minus 1 whole square minus 1 which is equal to 1 minus 1 which is again 0 and hence again we have p at minus 1 is equal to 0. Thus the value of the polynomial px at x is equal to 1 and x is equal to minus 1 are both 0. Thus we can say that yes x is equal to 1 and x is equal to minus 1 are both the zeros px which is equal to x square minus 1. So this completes the third part and now proceeding on to the next one where px is equal to x plus 1 into x minus 2 and the values of x where we have to verify whether it is a 0 of px or not are minus 1 and 2. So first let us substitute x is equal to minus 1 in the given polynomial then we have minus 1 into minus 1 minus 2 which is further equal to 0 into minus 3 and on multiplying 0 with any number the result is 0. So we have p at minus 1 is equal to 0. Now let us find the value of the polynomial px at x is equal to 2. So replacing x by 2 we have 2 plus 1 into 2 minus 2 which is equal to 2 plus 1 is 3 and 2 minus 2 is 0 and again on multiplying 0 with any number the result is 0. So we have p at 2 is equal to 0 and thus we can say that x is equal to minus 1 and x is equal to 2 are the zeros of the given polynomial which is px is equal to x plus 1 into x minus 2. So this completes the fourth part and now proceeding on to the fifth one which is px is equal to x square and x is equal to 0. So to check whether x is equal to 0 is a 0 of px or not let us replace the variable x by 0 and let us give 0 square 0 square is equal to 0. So we have p0 is equal to 0 and thus we have a key idea. We can say that here is x is equal to 0 is the 0 of the given polynomial which is px is equal to x square. So this completes the fifth part and now proceeding on to the sixth part which is px is equal to lx plus m where x is equal to minus m upon l. So to check whether x is equal to minus m upon l as a solution of px or not let us replace the variable x by minus m upon l and we have l on the right inside and minus m upon l plus m and now l is common in the numerator and denominator on cancelling we are left with minus m plus m which is equal to 0. So we have p at minus m plus l is equal to 0 thus we can say that yes x is equal to minus m upon l is a 0 of the given polynomial which is px is equal to lx plus m and now proceeding on to the seventh one which is px is equal to 3x square minus 1 and the values of x are x is equal to minus 1 upon root 3 2 upon root 3. So first let us check whether x is equal to minus 1 upon 3 is a 0 of px or not. So replacing x by minus 1 upon root 3 in the polynomial we have 3 into minus 1 upon root 3 whole square minus 1 which is further equal to 3 and minus 1 upon root 3 whole square is 1 upon 3 minus 1 3 cancels out with 3 we have 1 minus 1 which is equal to 0 thus we have p at minus 1 upon root 3 is equal to 0 and now let us find whether x is equal to 2 upon root 3 is a 0 of px or not. So again replacing x by 2 upon root 3 in the polynomial we have 3 into 2 upon root 3 whole square minus 1 which is equal to 3 2 whole square is 4 and root 3 whole square is 3 minus 1 now 3 cancels out with 3 and we are left with 4 minus 1 which is equal to 3 hence p at 2 upon root 3 is equal to 3 which is not equal to 0 hence we can say that x is equal to minus 1 upon root 3 is a 0 px which is equal to 3x square minus 1 and x is equal to 2 upon root over 3 is not a 0 of the same polynomial which is 3x square minus 1. So this completes the part and lastly let us check whether x is equal to half is a 0 of px or not. So on replacing x by half in the polynomial px let us check what the value of the polynomial comes. So we have 2 into minus half plus 1 and 2 cancels out with 2 we have 1 plus 1 which is equal to 2 hence p at half is equal to 2 which is not equal to 0 therefore we can say that no x is equal to half is not a 0 of the given polynomial which is 2x plus 1. So this completes the last part and thus let us write the answers of each of the problems answer of the first part is yes the given value of x is a 0 of the polynomial. Second one is no answer to the third part is yes answer to the fourth part is yes answer to the fifth part is also yes answer to the sixth part is also yes and answer to the seventh part is that x is equal to minus 1 upon root 3 is a 0 but 2 upon root 3 is not a 0 and the last one is so this completes the solution so did you find this problem interesting it was easy now all you have to do is to replace the variable by the given value in the polynomial if the value of the polynomial comes 0 and this implies the given value of the variable is the 0 if it does not come to 0 then it is not a 0. So hope you enjoyed this session take care and have a good day.