Using the graph and the factor theorem, the formula for the polynomial function of the least degree is:
[tex]p(x) = -\frac{1}{2}(x^3 - 7x - 6)[/tex]
The Factor Theorem states that a function that has zeros [tex]x_1, x_2, ..., x_n[/tex] can be written as:
[tex]p(x) = a(x - x_1)(x - x_2)...(x - x_n)[/tex]
In which a is the leading coefficient.
In the graph, the zeros are the values of x for which the x-axis is crossed, thus, they are:
[tex]x_1 = -2, x_2 = -1, x_3 = 3[/tex]
Then
[tex]p(x) = a(x + 2)(x + 1)(x - 3)[/tex]
[tex]p(x) = a(x^2 + 3x + 2)(x - 3)[/tex]
[tex]p(x) = a(x^3 - 7x - 6)[/tex]
The y-intercept is [tex]y = 3[/tex], which means that [tex]p(0) = 3[/tex], which is used to find a.
[tex]p(0) = a(0^3 - 7(0) - 6)[/tex]
[tex]p(0) = -6a[/tex]
Then
[tex]-6a = 3[/tex]
[tex]a = -\frac{3}{6}[/tex]
[tex]a = -\frac{1}{2}[/tex]
Then, the polynomial function is:
[tex]p(x) = -\frac{1}{2}(x^3 - 7x - 6)[/tex]
A similar problem is given at https://brainly.com/question/24380382
