Circle the correct letter: The equation x^3 - 30x^2 + 108x - 104 = 0 has a) No real roots; b) Exactly one real root; c) Three distinct real roots; d) A repeated root.

Firstly, the polynomial is a cubic, and so we know what its graph looks like, hence it must have at least one real root, and a) is false. Computing a root would be time consuming, so instead we adopt a different approach: differentiate the polynomial to get rid of the "-104", so that we get 3x^2 - 60x +108, and divide by 3 to get x^2 - 20x + 36, which quite obviously (by looking at the factors of 36) factorises into (x-18)(x-2). These are the points where the gradient is zero.

Plugging 2 into the polynomial, we obtain 0, and without even computing the value at x=18, we know the curve has a turning point at (2,0) and hence a repeated root, so that d) is the correct answer. Notice also that at first it looks like b) and d) may not be mutually exclusive, but by again keeping in mind the shape of the polynomial, and the fact that 2<18, we know that this turning point is a 'local' maximum, i.e. that the graph will curve down after x=2, ad then up from x=18, meaning that somewhere after x=18 there is another root, the value of which we are not interested in.