
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Question

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.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by stepSolved in 5 steps with 4 images

Knowledge Booster
Recommended textbooks for you
- Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY
- Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,

Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education

Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY

Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,

