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,...
Related questions
Question
100%
Help
![**Problem Statement:**
If \( x - 2 \) is a factor of \( x^4 - x^3 - mx^2 - 4 \), find the value of \( m \). Show all work to receive credit for your answer.
---
**Solution Details:**
When a polynomial \( f(x) \) has \( x - a \) as a factor, it means that \( f(a) = 0 \). For this problem, set \( a = 2 \) since \( x - 2 \) is the given factor.
1. **Substitute 2 into the polynomial:**
\[
f(x) = x^4 - x^3 - mx^2 - 4
\]
\[
f(2) = 2^4 - 2^3 - m(2^2) - 4
\]
2. **Calculate each term:**
\[
2^4 = 16, \quad 2^3 = 8, \quad 2^2 = 4
\]
3. **Substitute these values back into the equation:**
\[
f(2) = 16 - 8 - 4m - 4
\]
4. **Simplify the expression:**
\[
f(2) = 16 - 8 - 4m - 4 = 4 - 4m
\]
5. **Set the equation equal to zero (because \( f(2) = 0 \)):**
\[
4 - 4m = 0
\]
6. **Solve for \( m \):**
\[
4 = 4m
\]
\[
m = 1
\]
Therefore, the value of \( m \) is \( 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe4e0d9-a441-4076-9243-8e98c4a90bfe%2F13e0ab81-d561-479e-bac7-cee6f3b4f359%2Ffup8reg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If \( x - 2 \) is a factor of \( x^4 - x^3 - mx^2 - 4 \), find the value of \( m \). Show all work to receive credit for your answer.
---
**Solution Details:**
When a polynomial \( f(x) \) has \( x - a \) as a factor, it means that \( f(a) = 0 \). For this problem, set \( a = 2 \) since \( x - 2 \) is the given factor.
1. **Substitute 2 into the polynomial:**
\[
f(x) = x^4 - x^3 - mx^2 - 4
\]
\[
f(2) = 2^4 - 2^3 - m(2^2) - 4
\]
2. **Calculate each term:**
\[
2^4 = 16, \quad 2^3 = 8, \quad 2^2 = 4
\]
3. **Substitute these values back into the equation:**
\[
f(2) = 16 - 8 - 4m - 4
\]
4. **Simplify the expression:**
\[
f(2) = 16 - 8 - 4m - 4 = 4 - 4m
\]
5. **Set the equation equal to zero (because \( f(2) = 0 \)):**
\[
4 - 4m = 0
\]
6. **Solve for \( m \):**
\[
4 = 4m
\]
\[
m = 1
\]
Therefore, the value of \( m \) is \( 1 \).
Expert Solution
Step 1
Step by step
Solved in 3 steps with 4 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education