 Hi friends, welcome to this problem solving session on polynomials and the question is find the zeros of the quadratic polynomial x square plus 7x plus 12, this is a quadratic polynomial given and you have to verify the relation between the zeros and its coefficients, ok. So, how to solve such kind of problems? Now, let us say Bx is this given polynomial, you can clearly see it is a quadratic polynomial, it is mentioned also that is the quadratic polynomial, ok. And you have to first thing is find the zeros, what is the zero of a polynomial? So, you know a zero of a polynomial is that value of x which makes the polynomial equals to 0. So, Bx will become 0 for such values of x. So, those x values are called the zeros of the polynomial. So, how to find out the zero? So, equate the polynomial to zero and try to solve the quadratic equation you get, ok. So, now you have to find out those values of x for which the entire this value becomes 0, ok. So, we will adopt this writing the middle term and if you see I can write this as x square plus 3x plus 4x plus 12 is equal to 0, isn't it? Now, x can be taken common from the first two terms and it can be written as x into x plus 3 plus 4 times x plus 3 and this is equals to 0. So, hence it is x plus 3 times x plus 4 equals 0, right. Now, product of two factors is 0, we know how to deal with such cases. So, we know that either x plus 3 is 0, ok or x plus 4 is 0. So, this would imply x is equal to minus 3 or this will imply x is equal to minus 4. So, these are the two values of x or these are the zeros of the given polynomial, ok. Now, we have to verify the relationship between the zeros and the coefficients. So, we know that if alpha and beta are basically sum of zeros, sum of zeros is given by I think but minus b upon a, right. And what is b upon a? b upon a is nothing but that from the coefficients of a x square plus b x plus c, this is the polynomial. So, a is the coefficient of x square and b is the coefficient of x. So, sum of zeros should be equal to and our given polynomial was what x square plus 7x plus 12, isn't it. Now, clearly a is equal to 1 and b is equal to 7. So, hence sum of zeros in minus b by a is nothing but minus 7 upon 1 is equal to minus 7. Now, the zeros which we have found here minus 3 and minus 4 if you add both of them. So, minus 3 plus minus 4 is also equal to minus 7. So, if you see both are equal, isn't it. So, hence sum of 0 is minus b by a is verified, verified. Now, product of 0 product of product of zeros product of zeros for a quadratic polynomial is given by c upon a. Now, what is c here? c is clearly 12. So, c upon a is 12 upon 1 which is 12 and the product of zeros over minus minus 3 into minus 4 which is equal to 12 again. So, this is verified again. So, both sum of the roots and product of the roots are verified.