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
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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77a13500-3283-4aa2-b558-e9fbf21d692d%2Fbd9ee430-60a4-4ffb-ba82-8968ecb2c30b%2Fc5jsufn_processed.jpeg&w=3840&q=75)
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.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)