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

Tyra 2021-05-05 Answered

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?

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Expert Answer

hesgidiauE
Answered 2021-05-06 Author has 26077 answers

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

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