Find the polynomial function of degree 3 whose graph is shown in the figure.

To find a polynomial function of degree 3, we need to determine its general form and use the given points or characteristics of the graph to find its specific coefficients. The general form of a cubic polynomial is:

\[ f(x) = ax^3 + bx^2 + cx + d \]

If the graph is provided, we typically look for key features such as the x-intercepts (roots), y-intercept, and any given points or slopes.

### Steps to find the polynomial function:

1. **Identify the roots**: If the graph crosses the x-axis at \(x_1, x_2,\) and \(x_3\), then the polynomial can be written as:
   \[ f(x) = a(x - x_1)(x - x_2)(x - x_3) \]
   
2. **Determine the leading coefficient \(a\)**: This can be done using another point on the graph, often the y-intercept.

3. **Expand and simplify**: Once the roots and leading coefficient are determined, expand the polynomial and simplify to the standard form.

Since I do not have access to the specific graph, let's assume an example where the roots are \(x = -2, x = 1,\) and \(x = 3\), and the polynomial passes through the point \((0, 6)\).

1. **Write the polynomial with the roots**:
   \[ f(x) = a(x + 2)(x - 1)(x - 3) \]

2. **Find the leading coefficient \(a\)** using the y-intercept (0, 6):
   \[ 6 = a(0 + 2)(0 - 1)(0 - 3) \]
   \[ 6 = a \cdot 2 \cdot (-1) \cdot (-3) \]
   \[ 6 = a \cdot 6 \]
   \[ a = 1 \]

3. **Form the polynomial**:
   \[ f(x) = (x + 2)(x - 1)(x - 3) \]

4. **Expand the polynomial**:
   \[ f(x) = (x + 2)(x^2 - 4x + 3) \]
   \[ f(x) = x(x^2 - 4x + 3) + 2(x^2 - 4x + 3) \]
   \[ f(x) = x^3 - 4x^2 + 3x + 2x^2 - 8x + 6 \]
   \[ f(x) = x^3 - 2x^2 - 5x + 6 \]

So, the polynomial function of degree 3 is:
\[ f(x) = x^3 - 2x^2 - 5x + 6 \]

If you provide specific details or the graph, I can give a more accurate answer tailored to your exact problem.

Do you want any details or have any questions?

Here are some questions you can ask next:
1. How do you find the roots of a polynomial function?
2. How do you determine the turning points of a polynomial graph?
3. Can you explain the process of polynomial long division?
4. How can synthetic division be used with polynomials?
5. What are the applications of cubic polynomials in real life?

**Tip**: When finding polynomial functions from graphs, always note the x-intercepts and y-intercepts first as they provide critical information for constructing the function.

Learn how to find a polynomial function of degree 3 from its graph. This guide provides a detailed step-by-step solution, including identifying roots, determining coefficients, and expanding the polynomial. Suitable for Grades 10-12 students studying cubic polynomials.

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