Question

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

Polynomial graphs

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

2021-06-06

For the cubic function $$P(x)=a(x-\alpha)^{3}, a=0$$

If $$a>0$$, the graph is increasing by the increase of x.

If $$a<0$$, the graph is decreasing by the increase of$$x-\alpha$$

The geometrical significance of α, the graph intersects x-axis at the point $$(\alpha, 0)$$