Find a polynomial function of degree 4 with -4 as a zero of multiplicity 3 and 0 as a zero of multiplicity 1. The polynomial function in expanded form is f(x) = (Use 1 for the leading coefficient.)
Find a polynomial function of degree 4 with -4 as a zero of multiplicity 3 and 0 as a zero of multiplicity 1. The polynomial function in expanded form is f(x) = (Use 1 for the leading coefficient.)
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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![**Polynomial Functions: Problem Solving**
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**Problem Statement:**
Find a polynomial function of degree 4 with \( -4 \) as a zero of multiplicity 3 and \( 0 \) as a zero of multiplicity 1.
**Solution:**
The polynomial function in expanded form is \( f(x) = \) \[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \]
(Use 1 for the leading coefficient.)
**Details:**
- **Multiplicity of a zero**: Indicates the number of times a particular zero appears in the polynomial. For \( -4 \) to be a zero of the polynomial of multiplicity 3, the factor \( (x + 4) \) must be raised to the power of 3, i.e., \( (x + 4)^3 \).
- For \( 0 \) to be a zero of the polynomial of multiplicity 1, the factor is simply \( x \).
Therefore, the polynomial can be constructed by multiplying these factors together:
\[ f(x) = (x + 4)^3 \cdot x \]
To expand this into a polynomial:
\[ f(x) = x \cdot (x + 4)^3 \]
\[ = x \cdot (x^3 + 12x^2 + 48x + 64) \]
\[ = x^4 + 12x^3 + 48x^2 + 64x \]
Thus, the polynomial function in expanded form is:
\[ f(x) = x^4 + 12x^3 + 48x^2 + 64x \]
**Educational Note:**
When identifying the zeros and their multiplicities for constructing polynomial functions:
1. **Multiplicity** affects the shape of the graph at the zero:
- Even multiplicities touch the x-axis and turn around.
- Odd multiplicities cross the x-axis.
2. **Leading Coefficient**: By setting it to 1, the polynomial’s leading term is standardized, simplifying calculations and comparisons.
Understanding these principles helps in solving more complex polynomial equations and aids in graph interpretation.
---
This content is suitable for Algebra II or Pre-Calculus students learning about polynomial functions and their properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3230e3d-dd38-497e-8c1e-7c1c5570828c%2F704fd524-6884-4d99-93d0-6a8fdff4ead8%2Fq6tyimf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Polynomial Functions: Problem Solving**
---
**Problem Statement:**
Find a polynomial function of degree 4 with \( -4 \) as a zero of multiplicity 3 and \( 0 \) as a zero of multiplicity 1.
**Solution:**
The polynomial function in expanded form is \( f(x) = \) \[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \]
(Use 1 for the leading coefficient.)
**Details:**
- **Multiplicity of a zero**: Indicates the number of times a particular zero appears in the polynomial. For \( -4 \) to be a zero of the polynomial of multiplicity 3, the factor \( (x + 4) \) must be raised to the power of 3, i.e., \( (x + 4)^3 \).
- For \( 0 \) to be a zero of the polynomial of multiplicity 1, the factor is simply \( x \).
Therefore, the polynomial can be constructed by multiplying these factors together:
\[ f(x) = (x + 4)^3 \cdot x \]
To expand this into a polynomial:
\[ f(x) = x \cdot (x + 4)^3 \]
\[ = x \cdot (x^3 + 12x^2 + 48x + 64) \]
\[ = x^4 + 12x^3 + 48x^2 + 64x \]
Thus, the polynomial function in expanded form is:
\[ f(x) = x^4 + 12x^3 + 48x^2 + 64x \]
**Educational Note:**
When identifying the zeros and their multiplicities for constructing polynomial functions:
1. **Multiplicity** affects the shape of the graph at the zero:
- Even multiplicities touch the x-axis and turn around.
- Odd multiplicities cross the x-axis.
2. **Leading Coefficient**: By setting it to 1, the polynomial’s leading term is standardized, simplifying calculations and comparisons.
Understanding these principles helps in solving more complex polynomial equations and aids in graph interpretation.
---
This content is suitable for Algebra II or Pre-Calculus students learning about polynomial functions and their properties.
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