To find the equation of a degree 5 polynomial function given the described graph, we can follow these steps:

Step 1: Identify the roots

From the description, we know the polynomial passes through the points $(0, 32)$, $(2, 0)$, and $(2.5, -2)$. Specifically, we can identify the roots of the function where the graph crosses the x-axis.

• The root at $x = 2$ has multiplicity 2 because the graph touches the x-axis at $x = 2$ and doesn't cross it (it has a local minimum there).
• The root at $x = -2$ is also a root since the graph touches the x-axis there, suggesting multiplicity 2.

Thus, the roots of the polynomial are:

• $x = -2$ with multiplicity 2.
• $x = 2$ with multiplicity 2.
• There is another root near $x = -0.4$, as we can see from the description of the graph reaching a local maximum there. We can approximate it as $x = -0.4$, suggesting it has multiplicity 1.

Step 2: Construct the polynomial in factored form

We now know the polynomial must include these factors:

• A factor of $(x + 2)^2$ for the root at $x = -2$ (multiplicity 2).
• A factor of $(x - 2)^2$ for the root at $x = 2$ (multiplicity 2).
• A factor of $(x + 0.4)$ for the root near $x = -0.4$.

Therefore, the polynomial can be written as: $f(x) = a(x + 2)^2(x - 2)^2(x + 0.4)$ where $a$ is a constant that will adjust the vertical stretch/compression.

Step 3: Use known points to find $a$

We are given that the graph passes through the point $(-2.5, 24)$. This means that when $x = -2.5$, $f(x) = 24$. We substitute $x = -2.5$ into the equation to solve for $a$:

$24 = a(-2.5 + 2)^2(-2.5 - 2)^2(-2.5 + 0.4)$

Now, calculate the values: $(-2.5 + 2) = -0.5, \quad (-2.5 - 2) = -4.5, \quad (-2.5 + 0.4) = -2.1$ $24 = a(-0.5)^2(-4.5)^2(-2.1)$ $24 = a(0.25)(20.25)(-2.1)$ $24 = a(1.0640625)$ Now solve for $a$: $a = \frac{24}{1.0640625} \approx 22.57$

Step 4: Write the final equation

Now that we know $a$, the equation of the polynomial is: $f(x) = 22.57(x + 2)^2(x - 2)^2(x + 0.4)$

Final Answer:

The equation of the degree 5 polynomial in factored form is approximately: $f(x) = 22.57(x + 2)^2(x - 2)^2(x + 0.4)$

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Would you like more details on any part of the solution? Or would you like help with a different problem?

Here are 5 related questions to further explore:

1. How do you determine the multiplicity of a root from a graph?
2. How do you find the leading term of a polynomial from its factored form?
3. What is the role of the constant $a$ in the equation of a polynomial?
4. How do you graph a degree 5 polynomial given its roots and multiplicities?
5. How can you use synthetic division to find the equation of a polynomial?

Tip: Always pay attention to how the graph behaves at each root: if it crosses the x-axis, the multiplicity is odd; if it just touches the x-axis, the multiplicity is even!