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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
Question
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### Polynomial Graph Analysis

**Graph Description:**
The image displays a graph of a polynomial function. It is a smooth, continuous curve that intersects the x-axis at several points, indicating the zeros of the function.

**Key Points of the Graph:**
- The graph crosses the x-axis at approximately \( x = -1, 1, \) and \( 3 \).
- It exhibits a typical polynomial behavior with peaks and valleys, indicating local maxima and minima.

**Questions:**

a) **Determine the end behavior.**

b) **State the multiplicity for each of the zeroes of the graph.**

c) **Write the equation of the polynomial in factored form.**

### Explanation of End Behavior
In polynomial functions, end behavior is determined by the leading term. As \( x \) approaches positive or negative infinity, evaluate how the graph behaves based on the highest degree term.

### Multiplicity of Zeroes
The multiplicity of a zero refers to how many times a particular root appears in the polynomial. Observe whether the graph passes straight through or touches and turns at these intercepts to determine multiplicity.

### Factored Form
Factoring the polynomial gives insight into its roots and their multiplicities. Use the zeroes found from the graph to express the polynomial in terms of these roots.

This exercise is crucial for understanding polynomial functions, their graphs, and how they can be expressed in different formats.
Transcribed Image Text:### Polynomial Graph Analysis **Graph Description:** The image displays a graph of a polynomial function. It is a smooth, continuous curve that intersects the x-axis at several points, indicating the zeros of the function. **Key Points of the Graph:** - The graph crosses the x-axis at approximately \( x = -1, 1, \) and \( 3 \). - It exhibits a typical polynomial behavior with peaks and valleys, indicating local maxima and minima. **Questions:** a) **Determine the end behavior.** b) **State the multiplicity for each of the zeroes of the graph.** c) **Write the equation of the polynomial in factored form.** ### Explanation of End Behavior In polynomial functions, end behavior is determined by the leading term. As \( x \) approaches positive or negative infinity, evaluate how the graph behaves based on the highest degree term. ### Multiplicity of Zeroes The multiplicity of a zero refers to how many times a particular root appears in the polynomial. Observe whether the graph passes straight through or touches and turns at these intercepts to determine multiplicity. ### Factored Form Factoring the polynomial gives insight into its roots and their multiplicities. Use the zeroes found from the graph to express the polynomial in terms of these roots. This exercise is crucial for understanding polynomial functions, their graphs, and how they can be expressed in different formats.
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