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Given the graph of f(x), a third degree polynomial, which of the following factors of f(x) has a multiplicity greater than 1? -10 x + 4 x + 6 x - 3 x + 3 -5 -20- 15 10- 5 0 -5- -10 5 10

Given the graph of f(x), a third degree polynomial, which of the following factors of f(x) has a multiplicity greater than 1? -10 x + 4 x + 6 x - 3 x + 3 -5 -20- 15 10- 5 0 -5- -10 5 10

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 7E
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**Understanding Polynomials: Multiplicity of Roots** **Question:** Given the graph of \(f(x)\), a third-degree polynomial, which of the following factors of \(f(x)\) has a multiplicity greater than 1? **Graph Analysis:** The graph depicts a third-degree polynomial function \(f(x)\). The red curve intersects the x-axis at points where the function has roots. These points are roughly located at \(x = -6\), \(x = -4\), and \(x = 3\). From the provided graph: - The curve crosses the x-axis at \(x = -6\) and \(x = 3\). This indicates these points are single roots (multiplicity of 1). - The curve touches the x-axis at \(x = -4\) and turns around, which indicates that \(x = -4\) is a root with a multiplicity greater than 1. In polynomial terminology, this is known as a double root or having even multiplicity. **Answer Choices:** 1. \(x + 4\) 2. \(x + 6\) 3. \(x - 3\) 4. \(x + 3\) **Answer:** The correct factor of \(f(x)\) with a multiplicity greater than 1 is \(x + 4\). **Explanation:** The factor \(x + 4\) corresponds to the root at \(x = -4\), where the polynomial graph touches the x-axis and turns around, indicating a root with multiplicity greater than 1. The other factors correspond to roots where the curve crosses the x-axis, indicating a multiplicity of exactly 1. By understanding the behavior of polynomial graphs and their intersections with the x-axis, we can determine the multiplicities of their roots. This knowledge is foundational for solving polynomial equations and analyzing their properties.
Transcribed Image Text:**Understanding Polynomials: Multiplicity of Roots** **Question:** Given the graph of \(f(x)\), a third-degree polynomial, which of the following factors of \(f(x)\) has a multiplicity greater than 1? **Graph Analysis:** The graph depicts a third-degree polynomial function \(f(x)\). The red curve intersects the x-axis at points where the function has roots. These points are roughly located at \(x = -6\), \(x = -4\), and \(x = 3\). From the provided graph: - The curve crosses the x-axis at \(x = -6\) and \(x = 3\). This indicates these points are single roots (multiplicity of 1). - The curve touches the x-axis at \(x = -4\) and turns around, which indicates that \(x = -4\) is a root with a multiplicity greater than 1. In polynomial terminology, this is known as a double root or having even multiplicity. **Answer Choices:** 1. \(x + 4\) 2. \(x + 6\) 3. \(x - 3\) 4. \(x + 3\) **Answer:** The correct factor of \(f(x)\) with a multiplicity greater than 1 is \(x + 4\). **Explanation:** The factor \(x + 4\) corresponds to the root at \(x = -4\), where the polynomial graph touches the x-axis and turns around, indicating a root with multiplicity greater than 1. The other factors correspond to roots where the curve crosses the x-axis, indicating a multiplicity of exactly 1. By understanding the behavior of polynomial graphs and their intersections with the x-axis, we can determine the multiplicities of their roots. This knowledge is foundational for solving polynomial equations and analyzing their properties.
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