Use the Factor Theorem to determine if the binomials given are factors of f(x). THEN use these factors to finish factoring f(x). 4 ƒ (x) = x² − 6x³ + 7x² + 6x − 8 · given binomials: (x − 1) (x − 2)
Use the Factor Theorem to determine if the binomials given are factors of f(x). THEN use these factors to finish factoring f(x). 4 ƒ (x) = x² − 6x³ + 7x² + 6x − 8 · given binomials: (x − 1) (x − 2)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Using the Factor Theorem to Determine Binomial Factors**
To determine if the given binomials are factors of the polynomial \( f(x) \), use the Factor Theorem. The goal is to establish whether or not the binomials divide \( f(x) \) without a remainder.
**Polynomial Function:**
\[ f(x) = x^4 - 6x^3 + 7x^2 + 6x - 8 \]
**Given Binomials:**
1. \( (x - 1) \)
2. \( (x - 2) \)
**Steps:**
1. **Apply the Factor Theorem:**
- Substitute \( x = 1 \) into \( f(x) \) to check if \( f(1) = 0 \).
- Substitute \( x = 2 \) into \( f(x) \) to check if \( f(2) = 0 \).
2. **Determine factors based on results:**
- If \( f(1) = 0 \), \( (x - 1) \) is a factor.
- If \( f(2) = 0 \), \( (x - 2) \) is a factor.
3. **Complete the Factorization:**
- Use any confirmed factors to continue factoring \( f(x) \) until it is fully factored.
By following these steps, you can effectively use the Factor Theorem to verify and utilize binomial factors in polynomial expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27b17fd8-4e7e-48b6-9b45-28788af01d13%2Ff26f3654-2f62-475d-91ea-588d32fbb6ff%2Fubyei7k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using the Factor Theorem to Determine Binomial Factors**
To determine if the given binomials are factors of the polynomial \( f(x) \), use the Factor Theorem. The goal is to establish whether or not the binomials divide \( f(x) \) without a remainder.
**Polynomial Function:**
\[ f(x) = x^4 - 6x^3 + 7x^2 + 6x - 8 \]
**Given Binomials:**
1. \( (x - 1) \)
2. \( (x - 2) \)
**Steps:**
1. **Apply the Factor Theorem:**
- Substitute \( x = 1 \) into \( f(x) \) to check if \( f(1) = 0 \).
- Substitute \( x = 2 \) into \( f(x) \) to check if \( f(2) = 0 \).
2. **Determine factors based on results:**
- If \( f(1) = 0 \), \( (x - 1) \) is a factor.
- If \( f(2) = 0 \), \( (x - 2) \) is a factor.
3. **Complete the Factorization:**
- Use any confirmed factors to continue factoring \( f(x) \) until it is fully factored.
By following these steps, you can effectively use the Factor Theorem to verify and utilize binomial factors in polynomial expressions.
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