Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns aro each zero. f(x) = x3 + 2x2 -9x - 18 The zero(s) is/are -3,-2,3 (Type integers or decimals. Use a comma to separate answers as needed.) Determine the multiplicities of the zero(s), if they exist. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. O A. There are three zeros. The multiplicity of the smallest zero is The multiplicity of the largest zero is The multiplicity of the other zero is (Simplify your answers.) O B. There are two zeros. The multiplicity of the smallest zero is (Simplify your answers.) The multiplicity of the largest zero is O C. There is one zero. The multiplicity of the zero is (Simplify your answer.) Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the answer boxes within your choice. O A. The graph crosses the x-axis at x = (Type integers or decimals. Simplify your answers. Use a comma to separate answers as needed.) The graph touches the x-axis and turns around at x = O B. The graph crosses the x-axis at all zeros. C. The graph touches the x-axis and turns around at all zeros. (1,1) More

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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**Polynomial Function Zeros and Their Multiplicities**

_Task: Determine the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero._

**Given Polynomial Function:**
\[ f(x) = x^3 - 2x^2 - 9x - 18 \]

1. **Find the Zeros:**
   The zeros of the polynomial are \( x = \). 
   (Type integers or decimals. Use a comma to separate answers as needed.)
   \[= [-3, 2, 3] \]

2. **Determine the Multiplicities of the Zeros:** 
   (Note: If they exist, select the correct choices and simplify your answers.)
   - There are ___ zeros.
   - The multiplicity of the smallest zero is ___.
   - The multiplicity of the largest zero is ___.
   \\
   The multiplicity of the other zero is ___.

3. **Behavior of the Function at Each Zero:**
   Determine the behavior of the function at each zero below and, if necessary, fill in the answer boxes within your choice.
   - The graph crosses the x-axis at \( x = \) ___.
   - The graph touches the x-axis and turns around at \( x = \) __¬¬__.
   - (Type integers or decimals. Simplify your answers. Use a comma to separate answers as needed.)

### Explanation of Behavior at Zeros: 
- The graph crosses the x-axis at all zeros.
- The points where the graph touches the x-axis are critical in understanding how the polynomial behaves.

**Interactive Elements for Educational Website:**
- Multiple choice selection for identifying zeros and their multiplicities.
- Input boxes for students to manually enter the zeros and their properties.
- Options to select the behavior of the graph at each zero based on the multiplicity values.

### Graphical Representation:
- It is suggested to include an interactive graph that dynamically changes as students input the zeros and their multiplicities to visualize the polynomial behavior in real-time.

This structure ensures students understand not only how to find the zeros but also how these zeros affect the shape and behavior of the polynomial's graph.
Transcribed Image Text:**Polynomial Function Zeros and Their Multiplicities** _Task: Determine the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis or touches the x-axis and turns around at each zero._ **Given Polynomial Function:** \[ f(x) = x^3 - 2x^2 - 9x - 18 \] 1. **Find the Zeros:** The zeros of the polynomial are \( x = \). (Type integers or decimals. Use a comma to separate answers as needed.) \[= [-3, 2, 3] \] 2. **Determine the Multiplicities of the Zeros:** (Note: If they exist, select the correct choices and simplify your answers.) - There are ___ zeros. - The multiplicity of the smallest zero is ___. - The multiplicity of the largest zero is ___. \\ The multiplicity of the other zero is ___. 3. **Behavior of the Function at Each Zero:** Determine the behavior of the function at each zero below and, if necessary, fill in the answer boxes within your choice. - The graph crosses the x-axis at \( x = \) ___. - The graph touches the x-axis and turns around at \( x = \) __¬¬__. - (Type integers or decimals. Simplify your answers. Use a comma to separate answers as needed.) ### Explanation of Behavior at Zeros: - The graph crosses the x-axis at all zeros. - The points where the graph touches the x-axis are critical in understanding how the polynomial behaves. **Interactive Elements for Educational Website:** - Multiple choice selection for identifying zeros and their multiplicities. - Input boxes for students to manually enter the zeros and their properties. - Options to select the behavior of the graph at each zero based on the multiplicity values. ### Graphical Representation: - It is suggested to include an interactive graph that dynamically changes as students input the zeros and their multiplicities to visualize the polynomial behavior in real-time. This structure ensures students understand not only how to find the zeros but also how these zeros affect the shape and behavior of the polynomial's graph.
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