Let m and p be real numbers such that the polynomial f(x) = x^2 + mx + p has two distinct negative rational roots. Then the polynomial g(x) = x^2 - (m^2 - 2p)x + p^2 has distinct

Step-by-step solution

Let the roots of f(x) = x^2 + mx + p be and . Given that and are distinct negative rational numbers. From Vieta's formulas for f(x) : + = -m = p For the polynomial g(x) = x^2 - (m^2 - 2p)x + p^2 , the sum of the roots is m^2 - 2p and the product of the roots is p^2 . Expressing these in terms of and : m^2 - 2p = (-( + ))^2 - 2 = ^2 + ^2 + 2 - 2 = ^2 + ^2 p^2 = ( )^2 = ^2 ^2 Thus, the polynomial g(x) can be rewritten as: g(x) = x^2 - ( ^2 + ^2)x + ^2 ^2 = (x - ^2)(x - ^2) The roots of g(x) are ^2 and ^2 . Since and