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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![### Synthetic Division Example
1. Divide using synthetic division.
\[ \frac{2x^5 - 4x^4 + 3x^3 + 4x^2 - 3x + 4}{x + 2} \]
In this problem, we are given a polynomial \(2x^5 - 4x^4 + 3x^3 + 4x^2 - 3x + 4\) that we need to divide by \(x + 2\) using synthetic division.
Here are the steps:
1. **Identify the coefficients of the polynomial**:
- The polynomial \(2x^5 - 4x^4 + 3x^3 + 4x^2 - 3x + 4\) has these coefficients: 2, -4, 3, 4, -3, 4.
2. **Write the inverse of the divisor's coefficient**:
- The divisor is \(x + 2\). The inverse of 2 (which comes from \(x + 2 = 0\) solving to \(x = -2\)) is -2.
3. **Set up the synthetic division table**:
- Write -2 on the left side.
- Write the coefficients in a row: 2, -4, 3, 4, -3, 4.
4. **Perform the synthetic division**:
- Drop the first coefficient (2) straight down.
- Multiply -2 by 2, and write the result under the second coefficient.
- Add the second coefficient (-4) with this result (-4 + -4).
- Continue the process across all coefficients.
The synthetic division steps would look like this:
```
-2 | 2 -4 3 4 -3 4
| -4 16 -38 68 -130
------------------------
2 -8 19 -34 65 -126
```
### Explaining the Table:
- **First Row**: Coefficients of the polynomial.
- **Second Row**: The results from multiplying and adding each step.
- **Third Row**: The coefficients of the quotient polynomial.
### Quotient Polynomial
The result of the division is:
\[2x^4 - 8x^3 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbaa3e418-f7f0-46cd-8916-fbe0f2fd3c97%2Fad077817-a04a-4fa6-9471-8ffc1f92d5a1%2Fcems93.jpeg&w=3840&q=75)
Transcribed Image Text:### Synthetic Division Example
1. Divide using synthetic division.
\[ \frac{2x^5 - 4x^4 + 3x^3 + 4x^2 - 3x + 4}{x + 2} \]
In this problem, we are given a polynomial \(2x^5 - 4x^4 + 3x^3 + 4x^2 - 3x + 4\) that we need to divide by \(x + 2\) using synthetic division.
Here are the steps:
1. **Identify the coefficients of the polynomial**:
- The polynomial \(2x^5 - 4x^4 + 3x^3 + 4x^2 - 3x + 4\) has these coefficients: 2, -4, 3, 4, -3, 4.
2. **Write the inverse of the divisor's coefficient**:
- The divisor is \(x + 2\). The inverse of 2 (which comes from \(x + 2 = 0\) solving to \(x = -2\)) is -2.
3. **Set up the synthetic division table**:
- Write -2 on the left side.
- Write the coefficients in a row: 2, -4, 3, 4, -3, 4.
4. **Perform the synthetic division**:
- Drop the first coefficient (2) straight down.
- Multiply -2 by 2, and write the result under the second coefficient.
- Add the second coefficient (-4) with this result (-4 + -4).
- Continue the process across all coefficients.
The synthetic division steps would look like this:
```
-2 | 2 -4 3 4 -3 4
| -4 16 -38 68 -130
------------------------
2 -8 19 -34 65 -126
```
### Explaining the Table:
- **First Row**: Coefficients of the polynomial.
- **Second Row**: The results from multiplying and adding each step.
- **Third Row**: The coefficients of the quotient polynomial.
### Quotient Polynomial
The result of the division is:
\[2x^4 - 8x^3 +
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