Let S be the set of all positive integers m for which the equation x^4 - 10x^3 + mx^2 - 10x + 1 = 0 has exactly 4 distinct real roots. The sum of all elements in the set S is

Step-by-step solution

Given equation: x^4 - 10x^3 + mx^2 - 10x + 1 = 0 . Since x = 0 is not a root, divide the entire equation by x^2 : x^2 - 10x + m - 10 x + 1 x^2 = 0 (x^2 + 1 x^2 ) - 10 (x + 1 x ) + m = 0 Let t = x + 1 x . Then x^2 + 1 x^2 = t^2 - 2 . Substituting this into the equation, we get: (t^2 - 2) - 10t + m = 0 t^2 - 10t + (m - 2) = 0 For a given real t , the equation x + 1 x = t x^2 - tx + 1 = 0 has real roots if its discriminant D_x = t^2 - 4 > 0 , which means |t| > 2 . For x to have 4 distinct real roots, the quadratic in