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Given the function f(x)=x3-2x2-3x a) Apply the leading coefficient test. b) Find the zeros by factoring. c) Find additional points between the zeros. d) Sketch the graph. 5 m N N n 54 M 2 r r N m 4 n H N 3 4 n

Given the function f(x)=x3-2x2-3x a) Apply the leading coefficient test. b) Find the zeros by factoring. c) Find additional points between the zeros. d) Sketch the graph. 5 m N N n 54 M 2 r r N m 4 n H N 3 4 n

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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**Polynomial Analysis and Graphing** **Given the function** \( f(x) = x^3 - 2x^2 - 3x \). **Task:** a) **Apply the Leading Coefficient Test.** The leading term of the polynomial is \(x^3\), and its leading coefficient is positive. Therefore, as \(x\) approaches positive infinity, \(f(x)\) will also approach positive infinity. Conversely, as \(x\) approaches negative infinity, \(f(x)\) will approach negative infinity. b) **Find the Zeros by Factoring.** Factor the polynomial to find its zeros. \( f(x) = x(x^2 - 2x - 3) \) Further factor the quadratic: \( x(x - 3)(x + 1) \) Thus, the zeros are \(x = 0\), \(x = 3\), and \(x = -1\). c) **Find Additional Points Between the Zeros.** Choose values between the zeros to find additional points: For \(x = -2\): \( f(-2) = (-2)((-2)^2 - 2(-2) - 3) = -2(4 + 4 - 3) = -10 \) For \(x = 1\): \( f(1) = (1)((1)^2 - 2(1) - 3) = 1(1 - 2 - 3) = -4 \) d) **Sketch the Graph.** Plot the points and sketch a smooth curve through the zeros: \(x = -1\), \(x = 0\), and \(x = 3\). Include additional points \( (-2, -10) \) and \( (1, -4) \) to guide the shape of the graph. **Graph Explanation:** - The graph provided is a standard Cartesian coordinate plane with both x-axis and y-axis ranging from -5 to 5. - Use these axes to plot the zeros and other calculated points to accurately sketch the polynomial's behavior. This approach provides a complete analysis of the function \( f(x) = x^3 - 2x^2 - 3x \) with a focus on factoring and graphing.
Transcribed Image Text:**Polynomial Analysis and Graphing** **Given the function** \( f(x) = x^3 - 2x^2 - 3x \). **Task:** a) **Apply the Leading Coefficient Test.** The leading term of the polynomial is \(x^3\), and its leading coefficient is positive. Therefore, as \(x\) approaches positive infinity, \(f(x)\) will also approach positive infinity. Conversely, as \(x\) approaches negative infinity, \(f(x)\) will approach negative infinity. b) **Find the Zeros by Factoring.** Factor the polynomial to find its zeros. \( f(x) = x(x^2 - 2x - 3) \) Further factor the quadratic: \( x(x - 3)(x + 1) \) Thus, the zeros are \(x = 0\), \(x = 3\), and \(x = -1\). c) **Find Additional Points Between the Zeros.** Choose values between the zeros to find additional points: For \(x = -2\): \( f(-2) = (-2)((-2)^2 - 2(-2) - 3) = -2(4 + 4 - 3) = -10 \) For \(x = 1\): \( f(1) = (1)((1)^2 - 2(1) - 3) = 1(1 - 2 - 3) = -4 \) d) **Sketch the Graph.** Plot the points and sketch a smooth curve through the zeros: \(x = -1\), \(x = 0\), and \(x = 3\). Include additional points \( (-2, -10) \) and \( (1, -4) \) to guide the shape of the graph. **Graph Explanation:** - The graph provided is a standard Cartesian coordinate plane with both x-axis and y-axis ranging from -5 to 5. - Use these axes to plot the zeros and other calculated points to accurately sketch the polynomial's behavior. This approach provides a complete analysis of the function \( f(x) = x^3 - 2x^2 - 3x \) with a focus on factoring and graphing.
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