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
![**Problem: Polynomial Long Division**
Find the quotient and remainder using long division for:
\[
\frac{x^3 + 10x^2 + 24x + 20}{x + 2}
\]
**Answer:**
The quotient is: [Fill in the quotient]
The remainder is: [Fill in the remainder]
**Explanation:**
To solve this problem, apply polynomial long division. Follow these steps:
1. **Divide the first term of the dividend by the first term of the divisor:**
- \(x^3 \div x = x^2\)
2. **Multiply the entire divisor by this result and subtract:**
- Multiply: \(x^2 \times (x + 2) = x^3 + 2x^2\)
- Subtract:
\[
(x^3 + 10x^2 + 24x + 20) - (x^3 + 2x^2) = 8x^2 + 24x + 20
\]
3. **Repeat the process with the new polynomial:**
- \(8x^2 \div x = 8x\)
- Multiply: \(8x \times (x + 2) = 8x^2 + 16x\)
- Subtract:
\[
(8x^2 + 24x + 20) - (8x^2 + 16x) = 8x + 20
\]
4. **Continue the process:**
- \(8x \div x = 8\)
- Multiply: \(8 \times (x + 2) = 8x + 16\)
- Subtract:
\[
(8x + 20) - (8x + 16) = 4
\]
Thus, the quotient is \(x^2 + 8x + 8\) and the remainder is 4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe84762dd-baf9-4e17-b4ea-3980bb4862e9%2Fe150b5da-0f23-452f-a96a-394ffc0bf0ad%2Fcnpdffd.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Polynomial Long Division**
Find the quotient and remainder using long division for:
\[
\frac{x^3 + 10x^2 + 24x + 20}{x + 2}
\]
**Answer:**
The quotient is: [Fill in the quotient]
The remainder is: [Fill in the remainder]
**Explanation:**
To solve this problem, apply polynomial long division. Follow these steps:
1. **Divide the first term of the dividend by the first term of the divisor:**
- \(x^3 \div x = x^2\)
2. **Multiply the entire divisor by this result and subtract:**
- Multiply: \(x^2 \times (x + 2) = x^3 + 2x^2\)
- Subtract:
\[
(x^3 + 10x^2 + 24x + 20) - (x^3 + 2x^2) = 8x^2 + 24x + 20
\]
3. **Repeat the process with the new polynomial:**
- \(8x^2 \div x = 8x\)
- Multiply: \(8x \times (x + 2) = 8x^2 + 16x\)
- Subtract:
\[
(8x^2 + 24x + 20) - (8x^2 + 16x) = 8x + 20
\]
4. **Continue the process:**
- \(8x \div x = 8\)
- Multiply: \(8 \times (x + 2) = 8x + 16\)
- Subtract:
\[
(8x + 20) - (8x + 16) = 4
\]
Thus, the quotient is \(x^2 + 8x + 8\) and the remainder is 4.
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