The polynomial of degree 3, P(x), has a root of multiplicity 2 at x = 3 and a root of multiplicity 1 at 3. The y-intercept is y = 21.6. x= Find a formula for P(x). P(x) =
The polynomial of degree 3, P(x), has a root of multiplicity 2 at x = 3 and a root of multiplicity 1 at 3. The y-intercept is y = 21.6. x= Find a formula for P(x). P(x) =
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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![**Problem Statement: Polynomial Function with Given Roots and Multiplicity**
The polynomial of degree 3, \( P(x) \), has a root of multiplicity 2 at \( x = 3 \), and a root of multiplicity 1 at \( x = -3 \). The \( y \)-intercept is \( y = -21.6 \).
**Task:**
Find a formula for \( P(x) \).
**Given Information:**
- Degree of the polynomial: 3
- Roots and their multiplicities:
- \( x = 3 \) (Multiplicity: 2)
- \( x = -3 \) (Multiplicity: 1)
- \( y \)-intercept: \(-21.6\)
**Required Formula:**
\[ P(x) = \, \]
Fill in the blank with the equation of the polynomial.
**Explanation on how to find \( P(x) \):**
To find the polynomial \( P(x) \):
1. Identify the roots and their multiplicities. From the given data, we know:
- Root \( x = 3 \) has multiplicity 2, meaning it appears twice in the factorization.
- Root \( x = -3 \) has multiplicity 1, meaning it appears once in the factorization.
2. Construct the polynomial using these roots:
Since \( x = 3 \) has multiplicity 2, it contributes the factor \( (x-3)^2 \).
Since \( x = -3 \) has multiplicity 1, it contributes the factor \( (x+3) \).
3. Combine these factors to form the polynomial. Also, include a constant coefficient \( A \) to account for the \( y \)-intercept:
\[ P(x) = A (x - 3)^2 (x + 3) \]
4. Use the \( y \)-intercept to find the constant coefficient \( A \). At the \( y \)-intercept, \( x = 0 \) and \( P(0) = -21.6 \):
\[ -21.6 = A (0 - 3)^2 (0 + 3) \]
\[ -21.6 = A (-3)^2 (3) \]
\[ -21.6 = A (9)(3) \]
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11f50382-dc41-434c-a8e4-3cb9cbf912c9%2F50b93a56-7317-426a-b34f-b2802839226f%2Fd6m6t0f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Polynomial Function with Given Roots and Multiplicity**
The polynomial of degree 3, \( P(x) \), has a root of multiplicity 2 at \( x = 3 \), and a root of multiplicity 1 at \( x = -3 \). The \( y \)-intercept is \( y = -21.6 \).
**Task:**
Find a formula for \( P(x) \).
**Given Information:**
- Degree of the polynomial: 3
- Roots and their multiplicities:
- \( x = 3 \) (Multiplicity: 2)
- \( x = -3 \) (Multiplicity: 1)
- \( y \)-intercept: \(-21.6\)
**Required Formula:**
\[ P(x) = \, \]
Fill in the blank with the equation of the polynomial.
**Explanation on how to find \( P(x) \):**
To find the polynomial \( P(x) \):
1. Identify the roots and their multiplicities. From the given data, we know:
- Root \( x = 3 \) has multiplicity 2, meaning it appears twice in the factorization.
- Root \( x = -3 \) has multiplicity 1, meaning it appears once in the factorization.
2. Construct the polynomial using these roots:
Since \( x = 3 \) has multiplicity 2, it contributes the factor \( (x-3)^2 \).
Since \( x = -3 \) has multiplicity 1, it contributes the factor \( (x+3) \).
3. Combine these factors to form the polynomial. Also, include a constant coefficient \( A \) to account for the \( y \)-intercept:
\[ P(x) = A (x - 3)^2 (x + 3) \]
4. Use the \( y \)-intercept to find the constant coefficient \( A \). At the \( y \)-intercept, \( x = 0 \) and \( P(0) = -21.6 \):
\[ -21.6 = A (0 - 3)^2 (0 + 3) \]
\[ -21.6 = A (-3)^2 (3) \]
\[ -21.6 = A (9)(3) \]
\[
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