The equation \(x^3 - 30x^2 + 108x - 104 = 0\) has
no real roots;
exactly one real root;
three distinct real roots;
a repeated root.
Let \(f(x) = x^3 - 30x^2 + 108x - 104\).
By searching for small integer solutions, we find that \[f(2) = 8 - 120 + 216 - 104 = 0,\] so \((x - 2)\) is a factor of \(f(x)\). Then we can factorise completely to find \[f(x) = (x - 2)(x^2 - 28x + 52) = (x - 2)(x - 2)(x - 26),\] so \(f(x)\) has a repeated root and the answer is (d).
