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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Question
![**Problem Statement:**
Find a polynomial with integer coefficients that satisfies the given conditions.
**Conditions:**
- \( Q \) has degree 3 and zeros \( 5 \), \( 5i \), and \( -5i \).
**Solution:**
Q(x) = [Input box for the polynomial]
**Explanation:**
To find the polynomial \( Q(x) \), we need to use the fact that if a complex number is a zero of a polynomial with real coefficients, then its complex conjugate must also be a zero. Given the zeros \( 5 \), \( 5i \), and \( -5i \), the polynomial can be constructed as follows:
1. The polynomial can be represented by the factors associated with its zeros:
\[
Q(x) = (x - 5)(x - 5i)(x + 5i)
\]
2. Multiply out the complex conjugate pair:
\[
(x - 5i)(x + 5i) = x^2 + 25
\]
3. Then multiply the result by the remaining linear factor:
\[
Q(x) = (x - 5)(x^2 + 25)
\]
4. Expand:
\[
Q(x) = x(x^2 + 25) - 5(x^2 + 25) = x^3 + 25x - 5x^2 - 125
\]
5. Reorder the terms:
\[
Q(x) = x^3 - 5x^2 + 25x - 125
\]
Thus, the polynomial \( Q(x) \) is \( x^3 - 5x^2 + 25x - 125 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2cd8142-e63e-4851-bc3e-120e6814a336%2Fed7cb191-bd6f-4b28-a1be-f0e9996ca188%2F4etauza_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a polynomial with integer coefficients that satisfies the given conditions.
**Conditions:**
- \( Q \) has degree 3 and zeros \( 5 \), \( 5i \), and \( -5i \).
**Solution:**
Q(x) = [Input box for the polynomial]
**Explanation:**
To find the polynomial \( Q(x) \), we need to use the fact that if a complex number is a zero of a polynomial with real coefficients, then its complex conjugate must also be a zero. Given the zeros \( 5 \), \( 5i \), and \( -5i \), the polynomial can be constructed as follows:
1. The polynomial can be represented by the factors associated with its zeros:
\[
Q(x) = (x - 5)(x - 5i)(x + 5i)
\]
2. Multiply out the complex conjugate pair:
\[
(x - 5i)(x + 5i) = x^2 + 25
\]
3. Then multiply the result by the remaining linear factor:
\[
Q(x) = (x - 5)(x^2 + 25)
\]
4. Expand:
\[
Q(x) = x(x^2 + 25) - 5(x^2 + 25) = x^3 + 25x - 5x^2 - 125
\]
5. Reorder the terms:
\[
Q(x) = x^3 - 5x^2 + 25x - 125
\]
Thus, the polynomial \( Q(x) \) is \( x^3 - 5x^2 + 25x - 125 \).
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