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.
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
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Relevant Questions
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 \(\alpha, \beta, and\ \gamma ? \alpha,\beta,and\ \gamma\)?
asked 2021-05-30
Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: \(P(x)=a(x-\alpha)(x−\beta)(x−\gamma),a \neq 0\)
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: \(P(x)=a(x-\alpha)3,a \neq 0\)
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?
asked 2021-06-05
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?
asked 2020-12-06
For each polynomial function, one zero is given. Find all rational zeros and factor the polynomial. Then graph the function. \(f(x)=3x^{3}+x^{2}-10x-8\), zero:2
asked 2021-06-08
For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at \(x = -2,\) \(x = 1\), and \(x = 3\). y-intercept at \((0, -4)\).
asked 2021-06-20
For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at \(x = -5\), \(x =-2\), and \(x = 1\). y-intercept at (0, 6)
asked 2021-05-23
The graph of a polynomial function has the following characteristics: a) Its domain and range are the set of all real numbers. b) There are turning points at \(x = -2\), 0, and 3. a) Draw the graphs of two different polynomial functions that have these three characteristics. b) What additional characteristics would ensure that only one graph could be drawn?
