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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![### Question 9: Polynomial Division
**Task**: Find the quotient and remainder using long division for the polynomial expression:
\[
\frac{2x^3 - 14x^2 + 7x - 29}{2x^2 + 5}
\]
**Answer**:
- **The quotient is**: [Blank input field]
- **The remainder is**: \(2x + 6\)
Graphical/Diagram Explanation:
There's an interactive component indicating a multiple-choice question (Question 9) that requires students to identify the quotient and remainder when the polynomial \(2x^3 - 14x^2 + 7x - 29\) is divided by \(2x^2 + 5\) using the polynomial long division method. The frame provides blank fields for students to input their answer for the quotient, while the remainder has been already filled in as \(2x + 6\).
### Steps for Polynomial Long Division:
1. **Set up the division**: Write the dividend \(2x^3 - 14x^2 + 7x - 29\) and the divisor \(2x^2 + 5\) in long division format.
2. **Divide the first term**: Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(2x^2\) to get the first term of the quotient \(x\).
3. **Multiply and Subtract**:
- Multiply \(x\) by the divisor \(2x^2 + 5\) to get \(2x^3 + 5x\).
- Subtract \(2x^3 + 5x\) from \(2x^3 - 14x^2 + 7x - 29\) to get the new dividend of \(-14x^2 + 2x\).
4. **Repeat**: Continue this process with \(-14x^2 + 2x\) until the degree of the remaining polynomial is less than the degree of the divisor.
This procedure lets you clearly see how the quotient and remainder are determined for polynomial long division.
**Note**: This exercise illustrates polynomial long division, an important skill in algebra for simplifying expressions and solving polynomial equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F633100ea-0428-4ac9-9895-e60fbf23e986%2Fe6c6d7d7-0050-4df5-a7d1-e24a0f922146%2Fsgcl4zc.png&w=3840&q=75)
Transcribed Image Text:### Question 9: Polynomial Division
**Task**: Find the quotient and remainder using long division for the polynomial expression:
\[
\frac{2x^3 - 14x^2 + 7x - 29}{2x^2 + 5}
\]
**Answer**:
- **The quotient is**: [Blank input field]
- **The remainder is**: \(2x + 6\)
Graphical/Diagram Explanation:
There's an interactive component indicating a multiple-choice question (Question 9) that requires students to identify the quotient and remainder when the polynomial \(2x^3 - 14x^2 + 7x - 29\) is divided by \(2x^2 + 5\) using the polynomial long division method. The frame provides blank fields for students to input their answer for the quotient, while the remainder has been already filled in as \(2x + 6\).
### Steps for Polynomial Long Division:
1. **Set up the division**: Write the dividend \(2x^3 - 14x^2 + 7x - 29\) and the divisor \(2x^2 + 5\) in long division format.
2. **Divide the first term**: Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(2x^2\) to get the first term of the quotient \(x\).
3. **Multiply and Subtract**:
- Multiply \(x\) by the divisor \(2x^2 + 5\) to get \(2x^3 + 5x\).
- Subtract \(2x^3 + 5x\) from \(2x^3 - 14x^2 + 7x - 29\) to get the new dividend of \(-14x^2 + 2x\).
4. **Repeat**: Continue this process with \(-14x^2 + 2x\) until the degree of the remaining polynomial is less than the degree of the divisor.
This procedure lets you clearly see how the quotient and remainder are determined for polynomial long division.
**Note**: This exercise illustrates polynomial long division, an important skill in algebra for simplifying expressions and solving polynomial equations.
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