use the graph of the polynomial function and the given point to write the factored form of a polynomial…

Explanation:

Step1: Identify the x-intercepts from the graph

The graph crosses the x-axis at $x = -6$ and $x = 4$. It touches the x-axis and turns around at $x = -3$. $$x = -6, x = -3, x = 4$$

Step2: Determine the multiplicity of each root

Roots where the graph crosses have odd multiplicity (1), and roots where it touches have even multiplicity (2). $$(x + 6)^1, (x + 3)^2, (x - 4)^1$$

Step3: Write the general factored form

The polynomial function $f(x)$ with leading coefficient $a$ is: $$f(x) = a(x + 6)(x + 3)^2(x - 4)$$

Step4: Use the given point to find $a$

Substitute the point $(5, 9)$ into the equation: $$9 = a(5 + 6)(5 + 3)^2(5 - 4)$$

Step5: Solve for the leading coefficient $a$

$$9 = a(11)(8)^2(1) \implies 9 = a(11)(64) \implies 9 = 704a \implies a = \frac{9}{704}$$

Step6: Write the final polynomial function

Substitute $a$ back into the factored form equation. $$f(x) = \frac{9}{704}(x + 6)(x + 3)^2(x - 4)$$

Answer:

$f(x) = \frac{9}{704}(x + 6)(x + 3)^2(x - 4)$