If the equation x^4 - 4x^3 + ax^2 + bx + 1 = 0 has four positive roots, then the value of (a+b) is:

Step-by-step solution

Let the four positive roots be , , , . Sum of roots = + + + = 4 . Product of roots = = 1 . A.M. of roots = 4 4 = 1 . G.M. of roots = [4] 1 = 1 . Since A.M. = G.M., all roots must be equal to 1. i.e., = = = = 1 . So the equation is (x - 1)^4 = 0 . Expanding (x - 1)^4 gives x^4 - 4x^3 + 6x^2 - 4x + 1 = 0 . Comparing with x^4 - 4x^3 + ax^2 + bx + 1 = 0 , we get a = 6 and b = -4 . Therefore, a + b = 6 - 4 = 2 .