The sum and product of the zeros of a quadratic polynomial are - 1 2 and - 3 , respectively. What is the quadratic polynomial.

Let S and P denotes the sum and product of the zeros of the polynomial. Given that, Sum of zeros of the quadratic polynomial, S = - 1 2 Product of the zeros of the quadratic polynomial, P = - 3 Then the required polynomial g ( x ) is given by, g ( x ) = k x 2 - S x + P = k x 2 - - 1 2 x + ( - 3 ) = k x 2 + 1 2 x - 3 Hence, the quadratic polynomial is g ( x ) = k x 2 + x 2 - 3 , where k is any non-zero real number.

Write the family of quadratic polynomials having - 1 4 and 1 as it’s zeros.

The given zeros are - 1 4 and 1 . Therefore, ( x + 1 4 ) and ( x - 1 ) are the factors of the given polynomial So, ( x + 1 4 ) ( x - 1 ) = x 2 + 1 4 x - x - 1 4 = x 2 - 3 4 x - 1 4 Thus, the family of the quadratic polynomials is f x = k ( x 2 - 3 4 x - 1 4 ) , where k is any non-zero real number.