Determine if the given binomials are factors of the function. f(x) = 4x 5 - - 9x 4 3 + 39x + 24x 2 + 75x + 63; (4x + 3), (x - 1)
Determine if the given binomials are factors of the function. f(x) = 4x 5 - - 9x 4 3 + 39x + 24x 2 + 75x + 63; (4x + 3), (x - 1)
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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![**Determine if the given binomials are factors of the function.**
\[ f(x) = 4x^5 - 9x^4 + 39x^3 + 24x^2 + 75x + 63 \]
Determine if the binomials \( (4x + 3) \) and \( (x - 1) \) are factors of the function.
**Explanation:**
To determine if a binomial is a factor of a given polynomial function, use the Factor Theorem. According to this theorem, a binomial \( (x - c) \) is a factor of the polynomial \( f(x) \) if and only if \( f(c) = 0 \).
For the given binomials:
1. **Binomial \( (4x + 3) \):**
- Rewrite the binomial in the form \( (x - c) \) by solving \( 4x + 3 = 0 \), giving \( x = -\frac{3}{4} \).
- Substitute \( x = -\frac{3}{4} \) into the function \( f(x) \) to check if \( f\left(-\frac{3}{4}\right) = 0 \).
2. **Binomial \( (x - 1) \):**
- The binomial is already in the form \( (x - c) \), with \( c = 1 \).
- Substitute \( x = 1 \) into the function \( f(x) \) to check if \( f(1) = 0 \).
This process will determine if each binomial is a factor of the polynomial function \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F470394cf-97a2-4b61-9505-1db63f7a9e6d%2F5582f170-aeba-414d-95b7-2b558cf6bace%2Fxg87gc5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine if the given binomials are factors of the function.**
\[ f(x) = 4x^5 - 9x^4 + 39x^3 + 24x^2 + 75x + 63 \]
Determine if the binomials \( (4x + 3) \) and \( (x - 1) \) are factors of the function.
**Explanation:**
To determine if a binomial is a factor of a given polynomial function, use the Factor Theorem. According to this theorem, a binomial \( (x - c) \) is a factor of the polynomial \( f(x) \) if and only if \( f(c) = 0 \).
For the given binomials:
1. **Binomial \( (4x + 3) \):**
- Rewrite the binomial in the form \( (x - c) \) by solving \( 4x + 3 = 0 \), giving \( x = -\frac{3}{4} \).
- Substitute \( x = -\frac{3}{4} \) into the function \( f(x) \) to check if \( f\left(-\frac{3}{4}\right) = 0 \).
2. **Binomial \( (x - 1) \):**
- The binomial is already in the form \( (x - c) \), with \( c = 1 \).
- Substitute \( x = 1 \) into the function \( f(x) \) to check if \( f(1) = 0 \).
This process will determine if each binomial is a factor of the polynomial function \( f(x) \).
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