# 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: $$\displaystyle{P}{\left({x}\right)}={a}{\left({x}−α\right)}{\left({x}−β\right)}{\left({x}−γ\right)},{a}≠{0}$$
Type 2: Two real zeros, one repeated: $$\displaystyle{P}{\left({x}\right)}={a}{\left({x}−α\right)}{2}{\left({x}−β\right)},{a}≠{0}$$
Type 3: One real zero repeated three times: $$\displaystyle{P}{\left({x}\right)}={a}{\left({x}−α\right)}{3},{a}≠{0}$$
Type 4: One real and two imaginary zeros: $$\displaystyle{P}{\left({x}\right)}={\left({x}−α\right)}{\left({a}{x}{2}+{b}{x}+{c}\right)},Δ={b}{2}−{4}{a}{c}{<}{0},{a}≠{0}$$
Experiment with the graphs of Type 1 cubics. Clearly state the effect of changing both the size and sign of a. What is the geometrical significance of $$\alpha, \beta, and\ \gamma ? \alpha,\beta,and\ \gamma$$?

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The size of ¢ controls the vertical stretch or shrink of the graph Changing the sign of « reflects the graph about x-axis $$\alpha−\beta$$ and $$\gamma$$ are the x-intercepts of the graph.