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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![### Algebraic Factorization Practice
**Question 18**
**Which expression is a factor of \(2x^2 + 7x - 15\)?**
- [ ] \(2x - 5\)
- [ ] \(x + 5\)
- [X] \(x - 3\)
- [ ] \(2x + 3\)
To solve this problem, we need to factorize the quadratic expression \(2x^2 + 7x - 15\).
**Step-by-Step Solution:**
1. **Rewrite the middle term \(7x\) using two numbers that multiply to \(2 \cdot (-15) = -30\) and add to \(7\).**
These numbers are \(10\) and \(-3\), since \(10 \times (-3) = -30\) and \(10 + (-3) = 7\).
2. **Split the middle term \(7x\) accordingly:**
\[
2x^2 + 10x - 3x - 15
\]
3. **Group the terms:**
\[
(2x^2 + 10x) - (3x + 15)
\]
4. **Factor out the common terms from each group:**
\[
2x(x + 5) - 3(x + 5)
\]
5. **Factor out the common binomial \((x + 5)\):**
\[
(2x - 3)(x + 5)
\]
As a result, the expression is factored into \((2x - 3)(x + 5)\). Therefore, the factor \((x - 3)\) is **not** a correct factor. This may indicate that an error was made during factoring; let us recheck the process.
The factorization seems accurate if revisited, and our answer could have been a check-mark error.
### Correct Factorization:
Given the conflicting previous steps and correct answer:
- Correctly determining one's factor error is essential for correct algebraic practices.
**Note:** Proper quadratic factorization enhances solving efficiency and verification skills.
For further problems and detailed algebraic concept discussions, please visit our provided educational sections.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddcc5a2c-dcf8-4371-93cb-1de5d0b4469f%2F3f3e3245-c571-41de-8e51-ce3fa02bfdcd%2F7nj6w3r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Algebraic Factorization Practice
**Question 18**
**Which expression is a factor of \(2x^2 + 7x - 15\)?**
- [ ] \(2x - 5\)
- [ ] \(x + 5\)
- [X] \(x - 3\)
- [ ] \(2x + 3\)
To solve this problem, we need to factorize the quadratic expression \(2x^2 + 7x - 15\).
**Step-by-Step Solution:**
1. **Rewrite the middle term \(7x\) using two numbers that multiply to \(2 \cdot (-15) = -30\) and add to \(7\).**
These numbers are \(10\) and \(-3\), since \(10 \times (-3) = -30\) and \(10 + (-3) = 7\).
2. **Split the middle term \(7x\) accordingly:**
\[
2x^2 + 10x - 3x - 15
\]
3. **Group the terms:**
\[
(2x^2 + 10x) - (3x + 15)
\]
4. **Factor out the common terms from each group:**
\[
2x(x + 5) - 3(x + 5)
\]
5. **Factor out the common binomial \((x + 5)\):**
\[
(2x - 3)(x + 5)
\]
As a result, the expression is factored into \((2x - 3)(x + 5)\). Therefore, the factor \((x - 3)\) is **not** a correct factor. This may indicate that an error was made during factoring; let us recheck the process.
The factorization seems accurate if revisited, and our answer could have been a check-mark error.
### Correct Factorization:
Given the conflicting previous steps and correct answer:
- Correctly determining one's factor error is essential for correct algebraic practices.
**Note:** Proper quadratic factorization enhances solving efficiency and verification skills.
For further problems and detailed algebraic concept discussions, please visit our provided educational sections.
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