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9-14- 8- 5949 + 3 2 1 -5 -4 -3 -2 -1 1 Bo B 2 3 4 5 OF 6 8 p

9-14- 8- 5949 + 3 2 1 -5 -4 -3 -2 -1 1 Bo B 2 3 4 5 OF 6 8 p

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 35E
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**Understanding Polynomial Functions** **Question:** What is the least possible degree of the polynomial graphed above? **Explanation:** The given image features a coordinate plane graph with the polynomial function plotted. The graph shows critical points where the slope changes, indicating the location of the polynomial's roots. **Graph Analysis:** - The graph crosses the x-axis at three points, implying there could be three real roots. - The polynomial has two turning points, indicating the locations where the graph changes direction from increasing to decreasing (or vice versa). **Key Points:** 1. **Turning Points and Degree:** - The number of turning points of a polynomial function provides insight into its degree. Specifically, a polynomial of degree \( n \) can have at most \( n-1 \) turning points. - Since this graph has two turning points, the minimum degree for the polynomial must be at least 3. This is because a polynomial of degree \( n \) will have \( n-1 \) turning points. Therefore, \( n-1 \geq 2 \) implies \( n \geq 3 \). 2. **Roots and Degree:** - It is also worth noting that a polynomial's degree is at least as large as the number of its roots. However, since multiple roots might coincide at turning points, we rely more on turning points for minimal degree calculation. Therefore, the least possible degree of the polynomial graphed above is **3**.
Transcribed Image Text:**Understanding Polynomial Functions** **Question:** What is the least possible degree of the polynomial graphed above? **Explanation:** The given image features a coordinate plane graph with the polynomial function plotted. The graph shows critical points where the slope changes, indicating the location of the polynomial's roots. **Graph Analysis:** - The graph crosses the x-axis at three points, implying there could be three real roots. - The polynomial has two turning points, indicating the locations where the graph changes direction from increasing to decreasing (or vice versa). **Key Points:** 1. **Turning Points and Degree:** - The number of turning points of a polynomial function provides insight into its degree. Specifically, a polynomial of degree \( n \) can have at most \( n-1 \) turning points. - Since this graph has two turning points, the minimum degree for the polynomial must be at least 3. This is because a polynomial of degree \( n \) will have \( n-1 \) turning points. Therefore, \( n-1 \geq 2 \) implies \( n \geq 3 \). 2. **Roots and Degree:** - It is also worth noting that a polynomial's degree is at least as large as the number of its roots. However, since multiple roots might coincide at turning points, we rely more on turning points for minimal degree calculation. Therefore, the least possible degree of the polynomial graphed above is **3**.
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