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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Use synthetic division to determine the quotient of (5x^4+2x^2-15x+10)/(x+2). Example of how to solve attached.
Is x+2 a factor of the dividend?
![## Polynomial Long Division Example
### Problem Statement
Divide the polynomial \(2x^3 - 3x^2 + 4x + 36\) by \(x + 2\).
### Step-by-Step Solution
1. **Set Up the Division**:
\(2x^3 - 3x^2 + 4x + 36\) is divided by \(x + 2\).
2. **Determine the Zero**:
\[
x + 2 = 0 \Rightarrow x = -2
\]
3. **Synthetic Division Setup**:
Use \(x = -2\) for synthetic division:
- Write the coefficients from the dividend: \(2, -3, 4, 36\).
4. **Perform Synthetic Division**:
- Bring down the leading coefficient: \(2\).
- Multiply and add successively:
- \(2 \cdot -2 = -4\), add to \(-3\) to get \(-7\).
- \(-7 \cdot -2 = 14\), add to \(4\) to get \(18\).
- \(18 \cdot -2 = -36\), add to \(36\) to get \(0\).
5. **Results**:
- Quotient: \(2x^2 - 7x + 18\)
- Remainder: \(0\)
### Conclusion
Since the remainder is 0, \(x + 2\) and the quotient \(2x^2 - 7x + 18\) are factors of the dividend, \(2x^3 - 3x^2 + 4x + 36\).
### Verification
The dividend can be factored and expressed as:
\[
(x + 2)(2x^2 - 7x + 18) = 2x^3 - 3x^2 + 4x + 36
\]
This confirms the division is correct.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb552885-4792-4772-8974-c2c6ec1fb51b%2Fae00a5d2-5ecf-4c18-91de-d27a17956ee2%2Fk67mlc_processed.png&w=3840&q=75)
Transcribed Image Text:## Polynomial Long Division Example
### Problem Statement
Divide the polynomial \(2x^3 - 3x^2 + 4x + 36\) by \(x + 2\).
### Step-by-Step Solution
1. **Set Up the Division**:
\(2x^3 - 3x^2 + 4x + 36\) is divided by \(x + 2\).
2. **Determine the Zero**:
\[
x + 2 = 0 \Rightarrow x = -2
\]
3. **Synthetic Division Setup**:
Use \(x = -2\) for synthetic division:
- Write the coefficients from the dividend: \(2, -3, 4, 36\).
4. **Perform Synthetic Division**:
- Bring down the leading coefficient: \(2\).
- Multiply and add successively:
- \(2 \cdot -2 = -4\), add to \(-3\) to get \(-7\).
- \(-7 \cdot -2 = 14\), add to \(4\) to get \(18\).
- \(18 \cdot -2 = -36\), add to \(36\) to get \(0\).
5. **Results**:
- Quotient: \(2x^2 - 7x + 18\)
- Remainder: \(0\)
### Conclusion
Since the remainder is 0, \(x + 2\) and the quotient \(2x^2 - 7x + 18\) are factors of the dividend, \(2x^3 - 3x^2 + 4x + 36\).
### Verification
The dividend can be factored and expressed as:
\[
(x + 2)(2x^2 - 7x + 18) = 2x^3 - 3x^2 + 4x + 36
\]
This confirms the division is correct.
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