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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Solve like example below. Polynomial long division.
![### Polynomial Long Division
This image illustrates the process of performing polynomial long division. The divisor is \(x - 1\) and the dividend is \(x^3 + 3x^2 - 2x + 6\).
**Steps in the Division Process:**
1. **Divide the first term:**
- Divide \(x^3\) by \(x\) to get \(x^2\).
- Write \(x^2\) above the division bar.
2. **Multiply and Subtract:**
- Multiply \(x^2\) by \(x - 1\) to get \(x^3 - x^2\).
- Subtract \(x^3 - x^2\) from \(x^3 + 3x^2\) to get \(4x^2 - 2x\).
3. **Repeat the process:**
- Divide \(4x^2\) by \(x\) to get \(4x\).
- Multiply \(4x\) by \(x - 1\) to get \(4x^2 - 4x\).
- Subtract \(4x^2 - 4x\) from \(4x^2 - 2x\) to get \(2x + 6\).
4. **Continue dividing:**
- Divide \(2x\) by \(x\) to get \(2\).
- Multiply \(2\) by \(x - 1\) to get \(2x - 2\).
- Subtract \(2x - 2\) from \(2x + 6\) to obtain the remainder \(8\).
**Result:**
The quotient is \(x^2 + 4x + 2\) with a remainder of \(8\). The complete division can be written as:
\[
x^3 + 3x^2 - 2x + 6 = (x - 1)(x^2 + 4x + 2) + 8
\]
This means the quotient is \(x^2 + 4x + 2 + \frac{8}{x-1}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2574792-c8b9-4dbb-8bcc-60e76e59b591%2F56eb38b2-2c6c-4053-b288-978a1cfb31f0%2Fcq00wano_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Long Division
This image illustrates the process of performing polynomial long division. The divisor is \(x - 1\) and the dividend is \(x^3 + 3x^2 - 2x + 6\).
**Steps in the Division Process:**
1. **Divide the first term:**
- Divide \(x^3\) by \(x\) to get \(x^2\).
- Write \(x^2\) above the division bar.
2. **Multiply and Subtract:**
- Multiply \(x^2\) by \(x - 1\) to get \(x^3 - x^2\).
- Subtract \(x^3 - x^2\) from \(x^3 + 3x^2\) to get \(4x^2 - 2x\).
3. **Repeat the process:**
- Divide \(4x^2\) by \(x\) to get \(4x\).
- Multiply \(4x\) by \(x - 1\) to get \(4x^2 - 4x\).
- Subtract \(4x^2 - 4x\) from \(4x^2 - 2x\) to get \(2x + 6\).
4. **Continue dividing:**
- Divide \(2x\) by \(x\) to get \(2\).
- Multiply \(2\) by \(x - 1\) to get \(2x - 2\).
- Subtract \(2x - 2\) from \(2x + 6\) to obtain the remainder \(8\).
**Result:**
The quotient is \(x^2 + 4x + 2\) with a remainder of \(8\). The complete division can be written as:
\[
x^3 + 3x^2 - 2x + 6 = (x - 1)(x^2 + 4x + 2) + 8
\]
This means the quotient is \(x^2 + 4x + 2 + \frac{8}{x-1}\).
![**Problem: Polynomial Long Division**
Find the quotient for
\[
\frac{2m^4 + 6m^3 - 15m^2 - 18m + 27}{m^2 - 3}
\]
using the long division algorithm.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2574792-c8b9-4dbb-8bcc-60e76e59b591%2F56eb38b2-2c6c-4053-b288-978a1cfb31f0%2Fgsrdunh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Polynomial Long Division**
Find the quotient for
\[
\frac{2m^4 + 6m^3 - 15m^2 - 18m + 27}{m^2 - 3}
\]
using the long division algorithm.
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