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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can someone please help with a-c? Thank you.
![### Polynomial Graph Analysis
#### Given the following graph:
A graph is provided that seems to represent a polynomial function. It plots points and includes an x- and y-axis labeled with integer values.
**Graph Details:**
- The graph intersects the x-axis at three points: \(x = -1\), \(x = 1\), and \(x = 3\).
- The y-axis is labeled at increments of 100, spanning from -500 to 100.
- The curve shows a typical polynomial behavior, dipping and rising through the x-axis at marked intervals.
#### Questions and Answers:
**a) Determine the end behavior.**
- The graph indicates that as \(x \to -\infty\), \(P(x) \to \infty\).
- As \(x \to \infty\), \(P(x) \to \infty\).
**b) State the multiplicity for each of the zeros of the graph.**
- Zero at \(x = 1\) has multiplicity 1.
- Zero at \(x = -1\) has multiplicity 1.
- Zero at \(x = 3\) has multiplicity 2.
**c) Write the equation of the polynomial in factored form.**
- The equation is expressed as:
\[
P(x) = c(x-1)(x+1)(x-3)^2
\]
By examining the graph's behavior and how it crosses or touches the x-axis, these characteristics and the polynomial are derived.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8beeba18-127d-4f79-b572-133114d91afd%2F64602ae4-40f9-41c3-bece-33fbb8126f21%2Fepqk9j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Polynomial Graph Analysis
#### Given the following graph:
A graph is provided that seems to represent a polynomial function. It plots points and includes an x- and y-axis labeled with integer values.
**Graph Details:**
- The graph intersects the x-axis at three points: \(x = -1\), \(x = 1\), and \(x = 3\).
- The y-axis is labeled at increments of 100, spanning from -500 to 100.
- The curve shows a typical polynomial behavior, dipping and rising through the x-axis at marked intervals.
#### Questions and Answers:
**a) Determine the end behavior.**
- The graph indicates that as \(x \to -\infty\), \(P(x) \to \infty\).
- As \(x \to \infty\), \(P(x) \to \infty\).
**b) State the multiplicity for each of the zeros of the graph.**
- Zero at \(x = 1\) has multiplicity 1.
- Zero at \(x = -1\) has multiplicity 1.
- Zero at \(x = 3\) has multiplicity 2.
**c) Write the equation of the polynomial in factored form.**
- The equation is expressed as:
\[
P(x) = c(x-1)(x+1)(x-3)^2
\]
By examining the graph's behavior and how it crosses or touches the x-axis, these characteristics and the polynomial are derived.
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