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: Binomial Factor Identification
Which binomial is a factor \( x^2 - 12x + 32 \)?
**Options:**
1. \( x - 2 \)
2. \( x - 1 \)
3. \( x - 8 \)
4. \( x - 12 \)
### Explanation:
The task requires determining which of the given binomials is a factor of the quadratic expression \( x^2 - 12x + 32 \). To find this, we can factorize the quadratic expression into two binomials and compare the solutions.
#### Factorization Process:
Given quadratic expression:
\( x^2 - 12x + 32 \)
1. Look for two numbers that multiply to 32 and add to -12.
2. The numbers -4 and -8 satisfy these conditions (-4 * -8 = 32 and -4 + -8 = -12).
Thus, the quadratic can be expressed as:
\( (x - 4)(x - 8) \)
Therefore, the factors of \( x^2 - 12x + 32 \) are \( (x - 4) \) and \( (x - 8) \).
#### Conclusion:
Among the given options, \( x - 8 \) is a factor of \( x^2 - 12x + 32 \).
Thus, the correct answer is:
\[ \text{Option 3: } x - 8 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1246267-81e7-461b-97b7-8e9aaefad2fd%2F0d07ad19-3e41-4a1e-a2d5-fdca1fb64e9e%2Fs4g8ehm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question: Binomial Factor Identification
Which binomial is a factor \( x^2 - 12x + 32 \)?
**Options:**
1. \( x - 2 \)
2. \( x - 1 \)
3. \( x - 8 \)
4. \( x - 12 \)
### Explanation:
The task requires determining which of the given binomials is a factor of the quadratic expression \( x^2 - 12x + 32 \). To find this, we can factorize the quadratic expression into two binomials and compare the solutions.
#### Factorization Process:
Given quadratic expression:
\( x^2 - 12x + 32 \)
1. Look for two numbers that multiply to 32 and add to -12.
2. The numbers -4 and -8 satisfy these conditions (-4 * -8 = 32 and -4 + -8 = -12).
Thus, the quadratic can be expressed as:
\( (x - 4)(x - 8) \)
Therefore, the factors of \( x^2 - 12x + 32 \) are \( (x - 4) \) and \( (x - 8) \).
#### Conclusion:
Among the given options, \( x - 8 \) is a factor of \( x^2 - 12x + 32 \).
Thus, the correct answer is:
\[ \text{Option 3: } x - 8 \]
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