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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![### Constructing a Polynomial with Given Zeros
To find a polynomial \( f(x) \) that meets the specified criteria, follow these steps:
**Problem Statement:**
- Write a polynomial \( f(x) \) that satisfies the given conditions.
- The polynomial is of degree 3 with integer coefficients.
- The zeros of the polynomial are \( -2i \) and \( \frac{9}{7} \).
### Detailed Solution:
#### 1. Understanding the Zeros
Given the zeros of the polynomial:
- \( -2i \)
- \( \frac{9}{7} \)
Since the polynomial must have integer coefficients, the complex zeros must occur in conjugate pairs (to ensure the imaginary parts cancel out when expanded).
Thus, the zeros of the polynomial are:
- \( -2i \) (given)
- \( 2i \) (the conjugate of \( -2i \))
- \( \frac{9}{7} \) (given)
#### 2. Forming Factors from Zeros
Using the zeros, we write the polynomial in its factored form:
\[ (x + 2i)(x - 2i)\left( x - \frac{9}{7} \right) \]
#### 3. Simplifying Factors with Complex Zeros
Recall that \( (x + 2i)(x - 2i) \) simplifies using the difference of squares:
\[ (x + 2i)(x - 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4 \]
#### 4. Combining All Factors
Combine the simplified factors:
\[ (x^2 + 4)\left( x - \frac{9}{7} \right) \]
#### 5. Ensuring Integer Coefficients
To achieve integer coefficients, we clear the fraction by multiplying through by 7:
\[ 7(x^2 + 4)\left( x - \frac{9}{7} \right) = 7(x^2 + 4) \left( \frac{7x - 9}{7} \right) \]
Simplify this to:
\[ 7(x^2 + 4) \left( 7x - 9 \right) \]
Expanding this:
\[ (7) \times (x^2 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd02c2397-8a4d-4107-81f2-96f803d8dc68%2F86be4da7-5638-4638-a3f7-3c48dc6f3eee%2Fkn4zp1h.jpeg&w=3840&q=75)
Transcribed Image Text:### Constructing a Polynomial with Given Zeros
To find a polynomial \( f(x) \) that meets the specified criteria, follow these steps:
**Problem Statement:**
- Write a polynomial \( f(x) \) that satisfies the given conditions.
- The polynomial is of degree 3 with integer coefficients.
- The zeros of the polynomial are \( -2i \) and \( \frac{9}{7} \).
### Detailed Solution:
#### 1. Understanding the Zeros
Given the zeros of the polynomial:
- \( -2i \)
- \( \frac{9}{7} \)
Since the polynomial must have integer coefficients, the complex zeros must occur in conjugate pairs (to ensure the imaginary parts cancel out when expanded).
Thus, the zeros of the polynomial are:
- \( -2i \) (given)
- \( 2i \) (the conjugate of \( -2i \))
- \( \frac{9}{7} \) (given)
#### 2. Forming Factors from Zeros
Using the zeros, we write the polynomial in its factored form:
\[ (x + 2i)(x - 2i)\left( x - \frac{9}{7} \right) \]
#### 3. Simplifying Factors with Complex Zeros
Recall that \( (x + 2i)(x - 2i) \) simplifies using the difference of squares:
\[ (x + 2i)(x - 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4 \]
#### 4. Combining All Factors
Combine the simplified factors:
\[ (x^2 + 4)\left( x - \frac{9}{7} \right) \]
#### 5. Ensuring Integer Coefficients
To achieve integer coefficients, we clear the fraction by multiplying through by 7:
\[ 7(x^2 + 4)\left( x - \frac{9}{7} \right) = 7(x^2 + 4) \left( \frac{7x - 9}{7} \right) \]
Simplify this to:
\[ 7(x^2 + 4) \left( 7x - 9 \right) \]
Expanding this:
\[ (7) \times (x^2 +
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