Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
Find the polynomial function that has the following zeros
![**Complex Roots and Polynomial Expansion**
**Problem:**
Given the roots 3 and 2i, fully expand the polynomial function.
**Solution Requirement:**
Find the polynomial function given these roots.
**Transcription:**
Polynomial Function: _______________
---
### Explanation:
When given roots such as 3 and 2i, a polynomial with real coefficients must also include the complex conjugate of any non-real root. Therefore, for root 2i, its conjugate -2i must also be included.
### Steps:
1. **Identify the roots:**
- Real root: \( x = 3 \)
- Complex roots: \( x = 2i \) and \( x = -2i \)
2. **Form factors from roots:**
- Real root => \( (x - 3) \)
- Complex roots => \( (x - 2i)(x + 2i) \)
3. **Expand the complex roots factor:**
- \( (x - 2i)(x + 2i) \) = \( x^2 + 4 \) (Using the difference of squares: \( a^2 - b^2 = (a - b)(a + b) \))
4. **Combine all factors to form the polynomial:**
- Multiply \( (x - 3) \) by \( (x^2 + 4) \)
5. **Fully expand the polynomial:**
\[
P(x) = (x - 3)(x^2 + 4) = x^3 + 4x - 3x^2 - 12
\]
Hence, the polynomial function is given by:
\[
P(x) = x^3 - 3x^2 + 4x - 12
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39dbe924-8f22-4d8c-a5f5-0fa8155d12a1%2Fad0b3907-14e2-4eb2-bf9f-138ca8cacdfa%2Fmffasr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Complex Roots and Polynomial Expansion**
**Problem:**
Given the roots 3 and 2i, fully expand the polynomial function.
**Solution Requirement:**
Find the polynomial function given these roots.
**Transcription:**
Polynomial Function: _______________
---
### Explanation:
When given roots such as 3 and 2i, a polynomial with real coefficients must also include the complex conjugate of any non-real root. Therefore, for root 2i, its conjugate -2i must also be included.
### Steps:
1. **Identify the roots:**
- Real root: \( x = 3 \)
- Complex roots: \( x = 2i \) and \( x = -2i \)
2. **Form factors from roots:**
- Real root => \( (x - 3) \)
- Complex roots => \( (x - 2i)(x + 2i) \)
3. **Expand the complex roots factor:**
- \( (x - 2i)(x + 2i) \) = \( x^2 + 4 \) (Using the difference of squares: \( a^2 - b^2 = (a - b)(a + b) \))
4. **Combine all factors to form the polynomial:**
- Multiply \( (x - 3) \) by \( (x^2 + 4) \)
5. **Fully expand the polynomial:**
\[
P(x) = (x - 3)(x^2 + 4) = x^3 + 4x - 3x^2 - 12
\]
Hence, the polynomial function is given by:
\[
P(x) = x^3 - 3x^2 + 4x - 12
\]
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 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning