Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
Graph the polynomial function,
![The given function is labeled as part b and is expressed as:
\[ f(x) = 5(x - 4)(x + 2)^2(x + 6) \]
Explanation:
This function represents a polynomial with multiple factors. Let's break it down:
1. **Factors**:
- \( (x - 4) \) is a linear factor with a root at \( x = 4 \).
- \( (x + 2)^2 \) is a quadratic factor and has a double root at \( x = -2 \).
- \( (x + 6) \) is another linear factor with a root at \( x = -6 \).
2. **Scalar Multiple**:
- The polynomial is multiplied by a constant, 5, which stretches or compresses the graph vertically.
3. **Roots**:
- The polynomial will touch or cross the x-axis at \( x = 4, -2, \) and \( -6 \).
4. **Behavior at Roots**:
- At \( x = 4 \) and \( x = -6 \), the graph will cross the x-axis since these roots have odd multiplicities.
- At \( x = -2 \), the graph will touch the x-axis and turn around because the root has an even multiplicity of 2.
This type of function is typically analyzed to determine intercepts, behavior at roots, and end behavior, which provides insight into the overall shape of the polynomial graph.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d5fb762-cce2-43c2-bb3c-934ffc87c93e%2F436911b3-2297-4b41-8921-7556031b546d%2Fdetapkf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The given function is labeled as part b and is expressed as:
\[ f(x) = 5(x - 4)(x + 2)^2(x + 6) \]
Explanation:
This function represents a polynomial with multiple factors. Let's break it down:
1. **Factors**:
- \( (x - 4) \) is a linear factor with a root at \( x = 4 \).
- \( (x + 2)^2 \) is a quadratic factor and has a double root at \( x = -2 \).
- \( (x + 6) \) is another linear factor with a root at \( x = -6 \).
2. **Scalar Multiple**:
- The polynomial is multiplied by a constant, 5, which stretches or compresses the graph vertically.
3. **Roots**:
- The polynomial will touch or cross the x-axis at \( x = 4, -2, \) and \( -6 \).
4. **Behavior at Roots**:
- At \( x = 4 \) and \( x = -6 \), the graph will cross the x-axis since these roots have odd multiplicities.
- At \( x = -2 \), the graph will touch the x-axis and turn around because the root has an even multiplicity of 2.
This type of function is typically analyzed to determine intercepts, behavior at roots, and end behavior, which provides insight into the overall shape of the polynomial graph.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education