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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Question
![**Problem Statement:**
Find the value of \( k \) so that the remainder is zero when dividing the polynomial \( (x^3 + 18x^2 + kx + 4) \) by \( (x + 2) \).
**Explanation:**
To solve this problem, we can use the Remainder Theorem. The theorem states that the remainder of the division of a polynomial \( f(x) \) by a linear divisor \( (x - a) \) is equal to \( f(a) \).
Here, the divisor is \( (x + 2) \), which can be written as \( (x - (-2)) \). Therefore, according to the Remainder Theorem, the remainder will be zero if \( f(-2) = 0 \).
**Steps:**
1. Substitute \( x = -2 \) into the polynomial \( f(x) = x^3 + 18x^2 + kx + 4 \).
2. Calculate the result:
\[
(-2)^3 + 18(-2)^2 + k(-2) + 4 = 0
\]
3. Simplify and solve the equation for \( k \):
\[
-8 + 18(4) - 2k + 4 = 0
\]
4. Combine like terms:
\[
-8 + 72 - 2k + 4 = 0
\]
5. \(-8 + 72 + 4 = 68\), so:
\[
68 - 2k = 0
\]
6. Solve for \( k \):
\[
2k = 68
\]
\[
k = 34
\]
Thus, the value of \( k \) is 34 for the remainder to be zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F470394cf-97a2-4b61-9505-1db63f7a9e6d%2Fcd79efc6-1cc5-49ce-b079-d9211c03cef7%2Fd1znfno_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the value of \( k \) so that the remainder is zero when dividing the polynomial \( (x^3 + 18x^2 + kx + 4) \) by \( (x + 2) \).
**Explanation:**
To solve this problem, we can use the Remainder Theorem. The theorem states that the remainder of the division of a polynomial \( f(x) \) by a linear divisor \( (x - a) \) is equal to \( f(a) \).
Here, the divisor is \( (x + 2) \), which can be written as \( (x - (-2)) \). Therefore, according to the Remainder Theorem, the remainder will be zero if \( f(-2) = 0 \).
**Steps:**
1. Substitute \( x = -2 \) into the polynomial \( f(x) = x^3 + 18x^2 + kx + 4 \).
2. Calculate the result:
\[
(-2)^3 + 18(-2)^2 + k(-2) + 4 = 0
\]
3. Simplify and solve the equation for \( k \):
\[
-8 + 18(4) - 2k + 4 = 0
\]
4. Combine like terms:
\[
-8 + 72 - 2k + 4 = 0
\]
5. \(-8 + 72 + 4 = 68\), so:
\[
68 - 2k = 0
\]
6. Solve for \( k \):
\[
2k = 68
\]
\[
k = 34
\]
Thus, the value of \( k \) is 34 for the remainder to be zero.
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