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
![**Title: Solving Polynomial Division Problem**
**Problem Statement:**
The polynomial \( x^4 + px^3 + qx^2 + rx + 6 \) is exactly divisible by each of \( (x - 1) \), \( (x - 2) \) and \( (x - 3) \).
Find the values of \( p \), \( q \), and \( r \).
**Explanation:**
In this problem, we are given a polynomial \( x^4 + px^3 + qx^2 + rx + 6 \) which is divisible by \( (x - 1) \), \( (x - 2) \), and \( (x - 3) \). This implies that the given polynomial has roots at \( x = 1 \), \( x = 2 \), and \( x = 3 \).
To find the values of \( p \), \( q \), and \( r \), we can set up the polynomial equation and apply the fact that when each of these values is substituted into the polynomial, it should satisfy the equation \( x^4 + px^3 + qx^2 + rx + 6 = 0 \).
We can write our equations based on roots as follows:
1. For \( x = 1 \):
\[
1^4 + p(1)^3 + q(1)^2 + r(1) + 6 = 0
\]
Simplifies to:
\[
1 + p + q + r + 6 = 0
\]
\[
p + q + r = -7
\]
2. For \( x = 2 \):
\[
2^4 + p(2)^3 + q(2)^2 + r(2) + 6 = 0
\]
Simplifies to:
\[
16 + 8p + 4q + 2r + 6 = 0
\]
\[
8p + 4q + 2r = -22
\]
\[
4p + 2q + r = -11
\]
3. For \( x = 3 \):
\[
3^4 + p(3)^3 + q(3)^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c46edd8-92d8-4db2-98e9-676dbbbf50aa%2F49e9c8ca-8af6-415a-acae-2cb801a12940%2Fwjwa38_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Polynomial Division Problem**
**Problem Statement:**
The polynomial \( x^4 + px^3 + qx^2 + rx + 6 \) is exactly divisible by each of \( (x - 1) \), \( (x - 2) \) and \( (x - 3) \).
Find the values of \( p \), \( q \), and \( r \).
**Explanation:**
In this problem, we are given a polynomial \( x^4 + px^3 + qx^2 + rx + 6 \) which is divisible by \( (x - 1) \), \( (x - 2) \), and \( (x - 3) \). This implies that the given polynomial has roots at \( x = 1 \), \( x = 2 \), and \( x = 3 \).
To find the values of \( p \), \( q \), and \( r \), we can set up the polynomial equation and apply the fact that when each of these values is substituted into the polynomial, it should satisfy the equation \( x^4 + px^3 + qx^2 + rx + 6 = 0 \).
We can write our equations based on roots as follows:
1. For \( x = 1 \):
\[
1^4 + p(1)^3 + q(1)^2 + r(1) + 6 = 0
\]
Simplifies to:
\[
1 + p + q + r + 6 = 0
\]
\[
p + q + r = -7
\]
2. For \( x = 2 \):
\[
2^4 + p(2)^3 + q(2)^2 + r(2) + 6 = 0
\]
Simplifies to:
\[
16 + 8p + 4q + 2r + 6 = 0
\]
\[
8p + 4q + 2r = -22
\]
\[
4p + 2q + r = -11
\]
3. For \( x = 3 \):
\[
3^4 + p(3)^3 + q(3)^
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