Find a formula for the given polynomial.

In this question, we have to identify the zeros of the polynomial, along with a point, and then we get that the formula for the polynomial is:

[tex]p(x) = -0.5(x^3 - x^2 + 6x)[/tex]

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Equation of a polynomial, according to it's zeros:

Given a polynomial f(x), this polynomial has roots such that it can be written as: , in which a is the leading coefficient.

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Identifying the zeros:

Given the graph, the zeros are the points where the graph crosses the x-axis. In this question, they are:

[tex]x_1 = -2, x_2 = 0, x_3 = 3[/tex]

Thus

[tex]p(x) = a(x - x_{1})(x - x_{2})(x-x_3)[/tex]

[tex]p(x) = a(x - (-2))(x - 0)(x-3)[/tex]

[tex]p(x) = ax(x+2)(x-3)[/tex]

[tex]p(x) = ax(x^2 - x + 6)[/tex]

[tex]p(x) = a(x^3 - x^2 + 6x)[/tex]

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Leading coefficient:

Passes through point (2,-8), that is, when [tex]x = 2, y = -8[/tex], which is used to find a. So

[tex]p(x) = a(x^3 - x^2 + 6x)[/tex]

[tex]-8 = a(2^3 - 2^2 + 6*2)[/tex]

[tex]16a = -8[/tex]

[tex]a = -\frac{8}{16} = -0.5[/tex]

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Considering the zeros and the leading coefficient, the formula is:

[tex]p(x) = -0.5(x^3 - x^2 + 6x)[/tex]

A similar problem is found at https://brainly.com/question/16078990

The formula that represents the polynomial in the figure is [tex]p(x) = x^{3}-x^{2}-6\cdot x[/tex].

Based on the Fundamental Theorem of Algebra, we understand that Polynomials with real Coefficient have at least one real Root and at most a number of Roots equal to its Grade. The Grade is the maximum exponent that Polynomial has and root is a point such that [tex]p(x) = 0[/tex]. By Algebra we understand that polynomial can be represented in this manner known as Factorized form:

[tex]p(x) = \Pi\limits_{i=0}^{n} (x-r_i)[/tex] (1)

Where:

[tex]n[/tex] - Grade of the polynomial.

[tex]i[/tex] - Index of the root binomial.

[tex]x[/tex] - Independent variable.

We notice that polynomials has three roots in [tex]x = -2[/tex], [tex]x = 0[/tex] and [tex]x = 3[/tex], having the following construction:

[tex]p(x) =(x+2)\cdot x \cdot (x-3)[/tex]

[tex]p(x) = (x^{2}+2\cdot x)\cdot (x-3)[/tex]

[tex]p(x) = x^{3}+2\cdot x^{2}-3\cdot x^{2}-6\cdot x[/tex]

[tex]p(x) = x^{3}-x^{2}-6\cdot x[/tex]

The formula that represents the polynomial in the figure is [tex]p(x) = x^{3}-x^{2}-6\cdot x[/tex].

Here is a question related to the determination polynomials: https://brainly.com/question/10241002

The entire graph of the function f is shown in the figure below.
Write the domain and range of f using interval notation.

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Yes, because both figures are rectangles and all rectangles are similar.
No, because the center of dilation is not at (0, 0).
No, because corresponding sides have different slopes.

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+
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Smallest
5.07
5.76
50.67
55.706
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For a hypothesis test of the claim that the mean amount of sleep for adults is less than 6 ​hours, technology output shows that the hypothesis test has power of 0.4272 of supporting the claim that μ<6 hours of sleep when the actual population mean is 4.0 hours of sleep. Interpret this value of the​ power, then identify the value of β and interpret that value.

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Suppose the national mean annual salary for a school administrator is $91,000 a year. A school official took a sample of 25 school administrators in the state of Ohio to learn about salaries in that state to see if they differed from the national average.
77,600 76,000 90,700 97,200 90,700
101,800 78,700 81,300 84,200 97,600
77,500 75,700 89,400 84,300 78,700
84,600 87,700 103,400 83,800 101,300
94,700 69,200 95,400 61,500 68,800
(a) Formulate hypotheses that can be used to determine whether the population mean annual administrator salary in Ohio differs from the national mean of $91,000.
H0: μ ≤ 91,000
Ha: μ > 91,000
H0: μ > 91,000
Ha: μ ≤ 91,000
H0: μ ≥ 91,000
Ha: μ < 91,000
H0: μ < 91,000
Ha: μ ≥ 91,000
H0: μ = 91,000
Ha: μ ≠ 91,000
(b) The sample data for 25 Ohio administrators is contained in the file named Administrator.
What is the test statistic for your hypothesis test in part (a)? (Round your answer to three decimal places.)
What is the p-value for your hypothesis test in part (a)? (Round your answer to four decimal places.)
p-value =
(c) At
α = 0.05,
can your null hypothesis be rejected? What is your conclusion?
Reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.Do not reject H0. We cannot conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary. Reject H0. We cannot conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.Do not reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.
(d) Repeat the preceding hypothesis test using the critical value approach.
State the null and alternative hypotheses.
H0: μ ≤ 91,000
Ha: μ > 91,000
H0: μ > 91,000
Ha: μ ≤ 91,000
H0: μ ≥ 91,000
Ha: μ < 91,000
H0: μ < 91,000
Ha: μ ≥ 91,000
H0: μ = 91,000
Ha: μ ≠ 91,000
Find the value of the test statistic. (Round your answer to three decimal places.)
State the critical values for the rejection rule. Use
α = 0.05.
(Round your answer to three decimal places. If the test is one-tailed, enter NONE for the unused tail.)
test statistic≤test statistic≥
State your conclusion.
Reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.Do not reject H0. We cannot conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary. Reject H0. We cannot conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.Do not reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.

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