When f(x) is divided by x - 2, the remainder is 0. Given ƒ(x) x³ + 6x² = - x - 30, which conclusion about f(x) is NOT true? f(2)= 0 x - 2 is a factor of f(x) No conclusion can be made about f(x) x = 0
When f(x) is divided by x - 2, the remainder is 0. Given ƒ(x) x³ + 6x² = - x - 30, which conclusion about f(x) is NOT true? f(2)= 0 x - 2 is a factor of f(x) No conclusion can be made about f(x) x = 0
Chapter8: Roots And Radicals
Section8.7: Use Radicals In Functions
Problem 408E: Explain why the process of finding the domain of a radical function with an even index is different...
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![**Question for Analysis on Polynomial Functions**
When \( f(x) \) is divided by \( x - 2 \), the remainder is 0. Given
\[ f(x) = x^3 + 6x^2 - x - 30 \]
which conclusion about \( f(x) \) is NOT true?
a) \( f(2) = 0 \)
b) \( x - 2 \) is a factor of \( f(x) \)
c) No conclusion can be made about \( f(x) \)
d) \( x = 0 \)
**Explanation:**
To determine which conclusion is NOT true regarding the polynomial function \( f(x) = x^3 + 6x^2 - x - 30 \), let's analyze each statement:
1. **a) \( f(2) = 0 \)**: Since \( f(x) \) is divided by \( x - 2 \) and the remainder is 0, by the Factor Theorem, \( x - 2 \) is a factor of \( f(x) \). Therefore, \( f(2) = 0 \) is true because when \( x = 2 \), it should satisfy the equation.
2. **b) \( x - 2 \) is a factor of \( f(x) \)**: As mentioned previously, by the Factor Theorem, if \( f(x) \) is divided by \( x - 2 \) and the remainder is zero, then \( x - 2 \) is indeed a factor of \( f(x) \). Hence, this statement is true.
3. **c) No conclusion can be made about \( f(x) \)**: This statement is false because we can draw conclusions about \( f(x) \) based on the given information (that \( x - 2 \) is a factor of \( f(x) \)).
4. **d) \( x = 0 \)**: The statement \( x = 0 \) is unrelated to the information provided about the factor \( x - 2 \) and the polynomial function \( f(x) \). Hence, this conclusion is not relevant and not true in the context.
Thus, the answer is option c) No conclusion can be made about \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3692e55-09d2-4c7f-90e9-0ed90d8b0dce%2Ff4b98b43-bc78-4a4b-86c0-73afd818e6b6%2F395hxrm_processed.png&w=3840&q=75)
Transcribed Image Text:**Question for Analysis on Polynomial Functions**
When \( f(x) \) is divided by \( x - 2 \), the remainder is 0. Given
\[ f(x) = x^3 + 6x^2 - x - 30 \]
which conclusion about \( f(x) \) is NOT true?
a) \( f(2) = 0 \)
b) \( x - 2 \) is a factor of \( f(x) \)
c) No conclusion can be made about \( f(x) \)
d) \( x = 0 \)
**Explanation:**
To determine which conclusion is NOT true regarding the polynomial function \( f(x) = x^3 + 6x^2 - x - 30 \), let's analyze each statement:
1. **a) \( f(2) = 0 \)**: Since \( f(x) \) is divided by \( x - 2 \) and the remainder is 0, by the Factor Theorem, \( x - 2 \) is a factor of \( f(x) \). Therefore, \( f(2) = 0 \) is true because when \( x = 2 \), it should satisfy the equation.
2. **b) \( x - 2 \) is a factor of \( f(x) \)**: As mentioned previously, by the Factor Theorem, if \( f(x) \) is divided by \( x - 2 \) and the remainder is zero, then \( x - 2 \) is indeed a factor of \( f(x) \). Hence, this statement is true.
3. **c) No conclusion can be made about \( f(x) \)**: This statement is false because we can draw conclusions about \( f(x) \) based on the given information (that \( x - 2 \) is a factor of \( f(x) \)).
4. **d) \( x = 0 \)**: The statement \( x = 0 \) is unrelated to the information provided about the factor \( x - 2 \) and the polynomial function \( f(x) \). Hence, this conclusion is not relevant and not true in the context.
Thus, the answer is option c) No conclusion can be made about \( f(x) \).
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