# Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: P(x)=a(x−α)(x−β)(x−γ),a≠0 Type 2: Two real zeros

Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: $P\left(x\right)=a\left(x-\alpha \right)\left(x-\beta \right)\left(x-\gamma \right),a\ne 0$
Type 2: Two real zeros, one repeated: $P\left(x\right)=a\left(x-\alpha \right)2\left(x-\beta \right),a\ne 0$
Type 3: One real zero repeated three times: $P\left(x\right)=a\left(x-\alpha \right)3,a\ne 0$

Type 4: One real and two imaginary zeros: $P\left(x\right)=\left(x-\alpha \right)\left(ax2+bx+c\right),\mathrm{△}=b2-4ac<0,a\ne 0$
Experiment with the graphs of Type 4 cubics. What is the geometrical significance of $\alpha \alpha$ and the quadratic factor which has imaginary zeros?

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For the graph of the function $P\left(x\right)=\left(x-\alpha \right)\left(a{x}^{2}+bx+c\right),\mathrm{△}={b}^{2}-4ac<0,a=/0$
There is only one x-intercept, $\left(\alpha ,0\right)$
The other zeros are imaginary.