Question

Given that f(x) = 8x^3 + 20x^2 + 2nx - n - 6
(i) Show that x = 1 is a solution to f(x/2) = 0 for all real values of n.
(ii) Find the range of values of n for which x = 1 is the only root to the equation x^3 + 5x^2 + nx - n - 6 = 0

Answer 1

This following should help:\[2 n x-n+8 x^3+20 x^2-6=(2 x-1) \left(n+4 x^2+12 x+6\right) \]