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
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asked 2021-06-28

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 \(\alpha, \beta, and\ \gamma ? \alpha,\beta,and\ \gamma\)?

Answers (1)

2021-06-29

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.

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