Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: \(P(x)=a(x-\alpha)(x−\beta)(x−\gamma),a \neq0\)
Type 2: Two real zeros, one repeated: \(P(x)=a(x−\alpha)2(x-\beta),a \neq 0\)
Type 3: One real zero repeated three times: \(P(x)=a(x-\alpha)3,a \neq 0\)
Type 4: One real and two imaginary zeros: \(P(x)=(x-\alpha)(ax2+bx+c),\triangle=b2−4ac<0,a \neq0\)
Experiment with the graphs of Type 4 cubics. What is the geometrical significance of \(\alpha \alpha\) and the quadratic factor which has imaginary zeros?