Find the degree, leading coefficients, and the maximum number of real zeros of the polynomial. f(x) = 3x6 + 6 + 4x³ - 4x5 Degree Leading Coefficient = Maximum number of real zeros = = =
Find the degree, leading coefficients, and the maximum number of real zeros of the polynomial. f(x) = 3x6 + 6 + 4x³ - 4x5 Degree Leading Coefficient = Maximum number of real zeros = = =
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...
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![### Understanding Polynomials: Analyzing Degree, Leading Coefficients, and Real Zeros
Consider the polynomial function:
\[ f(x) = -3x^6 + 6 + 4x^3 - 4x^5 \]
### Key Terms:
1. **Degree:** The degree of a polynomial is the highest power of the variable in the polynomial.
2. **Leading Coefficient:** The leading coefficient is the coefficient of the term with the highest degree.
3. **Maximum Number of Real Zeros:** The maximum number of real zeros of a polynomial is equal to its degree.
### Steps to Analyze the Given Polynomial:
1. **Identify the Degree:**
The polynomial is \( f(x) = -3x^6 + 6 + 4x^3 - 4x^5 \).
- The term with the highest power of \( x \) is \( -3x^6 \).
- Therefore, the degree of this polynomial is 6.
**Degree =** \( \boxed{6} \)
2. **Determine the Leading Coefficient:**
- The leading coefficient is the coefficient of the term with the highest degree.
- In this polynomial, the term with the highest degree is \( -3x^6 \), and its coefficient is -3.
**Leading Coefficient =** \( \boxed{-3} \)
3. **Find the Maximum Number of Real Zeros:**
- The maximum number of real zeros a polynomial can have is equal to its degree.
- Since the degree of this polynomial is 6, it can have at most 6 real zeros.
**Maximum number of real zeros =** \( \boxed{6} \)
These are the key characteristics of the given polynomial \( f(x) = -3x^6 + 6 + 4x^3 - 4x^5 \). Understanding these can help in further analyzing and graphing the polynomial function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d8b0e94-47ef-477e-9622-abc8b98247d8%2F7fddb525-a161-46fc-9540-2df11a1326a4%2F8s6mqkk_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Polynomials: Analyzing Degree, Leading Coefficients, and Real Zeros
Consider the polynomial function:
\[ f(x) = -3x^6 + 6 + 4x^3 - 4x^5 \]
### Key Terms:
1. **Degree:** The degree of a polynomial is the highest power of the variable in the polynomial.
2. **Leading Coefficient:** The leading coefficient is the coefficient of the term with the highest degree.
3. **Maximum Number of Real Zeros:** The maximum number of real zeros of a polynomial is equal to its degree.
### Steps to Analyze the Given Polynomial:
1. **Identify the Degree:**
The polynomial is \( f(x) = -3x^6 + 6 + 4x^3 - 4x^5 \).
- The term with the highest power of \( x \) is \( -3x^6 \).
- Therefore, the degree of this polynomial is 6.
**Degree =** \( \boxed{6} \)
2. **Determine the Leading Coefficient:**
- The leading coefficient is the coefficient of the term with the highest degree.
- In this polynomial, the term with the highest degree is \( -3x^6 \), and its coefficient is -3.
**Leading Coefficient =** \( \boxed{-3} \)
3. **Find the Maximum Number of Real Zeros:**
- The maximum number of real zeros a polynomial can have is equal to its degree.
- Since the degree of this polynomial is 6, it can have at most 6 real zeros.
**Maximum number of real zeros =** \( \boxed{6} \)
These are the key characteristics of the given polynomial \( f(x) = -3x^6 + 6 + 4x^3 - 4x^5 \). Understanding these can help in further analyzing and graphing the polynomial function.
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