Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![## Problem Statement
Find the value of \( k \) so that when \( x^3 + kx^2 - 6x + (7 - k) \) is divided by \( x - 3 \), the remainder is 0.
## Multiple Choice Options
- \( \textcircled{2} \)
- \( \textcircled{-1} \)
- \( \textcircled{9} \)
- \( \textcircled{-2} \)
## Explanation
To solve this, use the Remainder Theorem which states that if a polynomial \( f(x) \) is divided by \( x - c \), the remainder of this division is \( f(c) \).
Given the polynomial \( f(x) = x^3 + kx^2 - 6x + (7 - k) \), and knowing that it is divided by \( x - 3 \) with a remainder of zero, we can write:
\[ f(3) = 0 \]
So substitute \( x = 3 \) into the polynomial:
\[ f(3) = 3^3 + k(3^2) - 6(3) + (7 - k) \]
\[ 0 = 27 + 9k - 18 + (7 - k) \]
\[ 0 = 27 + 9k - 18 + 7 - k \]
\[ 0 = 16 + 8k \]
Solve for \( k \):
\[ 8k = -16 \]
\[ k = -2 \]
Thus, the value of \( k \) is \(\textcircled{-2}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3692e55-09d2-4c7f-90e9-0ed90d8b0dce%2Fe811012a-dc4a-472d-bbad-d752baa92c9b%2Fe2eehq_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement
Find the value of \( k \) so that when \( x^3 + kx^2 - 6x + (7 - k) \) is divided by \( x - 3 \), the remainder is 0.
## Multiple Choice Options
- \( \textcircled{2} \)
- \( \textcircled{-1} \)
- \( \textcircled{9} \)
- \( \textcircled{-2} \)
## Explanation
To solve this, use the Remainder Theorem which states that if a polynomial \( f(x) \) is divided by \( x - c \), the remainder of this division is \( f(c) \).
Given the polynomial \( f(x) = x^3 + kx^2 - 6x + (7 - k) \), and knowing that it is divided by \( x - 3 \) with a remainder of zero, we can write:
\[ f(3) = 0 \]
So substitute \( x = 3 \) into the polynomial:
\[ f(3) = 3^3 + k(3^2) - 6(3) + (7 - k) \]
\[ 0 = 27 + 9k - 18 + (7 - k) \]
\[ 0 = 27 + 9k - 18 + 7 - k \]
\[ 0 = 16 + 8k \]
Solve for \( k \):
\[ 8k = -16 \]
\[ k = -2 \]
Thus, the value of \( k \) is \(\textcircled{-2}\).
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