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8 -6 -4 2 0 --2- (1,0) Graph of f(x) -~ 2 (3, 0)

8 -6 -4 2 0 --2- (1,0) Graph of f(x) -~ 2 (3, 0)

Advanced Engineering Mathematics
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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### Graph Details The image displays the graph of a polynomial function \( f(x) \). The graph intersects the x-axis at three points: \( (-2.5, 0) \), \( (-1, 0) \), and \( (3, 0) \). These points are the roots of the polynomial. ### Questions and Options: Given the graph of the polynomial function \( f(x) \) above, which of the following statements must be true? Select all that apply. - [ ] If the function \( f(x) \) is divided by \( (2x+1) \) the remainder is 0 - [ ] \( (x-1) \) is a factor of \( f(x) \) - [ ] \( (x+3) \) is a factor of \( f(x) \) - [ ] When the function \( f(x) \) is divided by \( (x-3) \) the remainder is 0 - [ ] When \( f(x) \) is divided by \( (x+1) \) the remainder is 0 ### Explanation On an educational website, it is essential to analyze the roots of the polynomial based on where it crosses the x-axis. Each root corresponds to a factor of the polynomial: - The root at \( x = -2.5 \) suggests a factor can be derived, but it does not match any integer-based factor from standard polynomials. - The roots at \( x = -1 \) and \( x = 3 \) suggest the polynomial can be factored to include \( (x + 1) \) and \( (x - 3) \), respectively. Thus, statements involving these factors might apply.
Transcribed Image Text:### Graph Details The image displays the graph of a polynomial function \( f(x) \). The graph intersects the x-axis at three points: \( (-2.5, 0) \), \( (-1, 0) \), and \( (3, 0) \). These points are the roots of the polynomial. ### Questions and Options: Given the graph of the polynomial function \( f(x) \) above, which of the following statements must be true? Select all that apply. - [ ] If the function \( f(x) \) is divided by \( (2x+1) \) the remainder is 0 - [ ] \( (x-1) \) is a factor of \( f(x) \) - [ ] \( (x+3) \) is a factor of \( f(x) \) - [ ] When the function \( f(x) \) is divided by \( (x-3) \) the remainder is 0 - [ ] When \( f(x) \) is divided by \( (x+1) \) the remainder is 0 ### Explanation On an educational website, it is essential to analyze the roots of the polynomial based on where it crosses the x-axis. Each root corresponds to a factor of the polynomial: - The root at \( x = -2.5 \) suggests a factor can be derived, but it does not match any integer-based factor from standard polynomials. - The roots at \( x = -1 \) and \( x = 3 \) suggest the polynomial can be factored to include \( (x + 1) \) and \( (x - 3) \), respectively. Thus, statements involving these factors might apply.
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