Question
Determine if the following function is a polynomial function. If it is, identify the degree.
f(x)equals StartFraction x squared plus 6 Over x Superscript 6 EndFraction
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Part 1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
It is a polynomial. The degree of the polynomial is
enter your response here.
It is not a polynomial.
f(x)equals StartFraction x squared plus 6 Over x Superscript 6 EndFraction
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
It is a polynomial. The degree of the polynomial is
enter your response here.
It is not a polynomial.
Answers
Answered by
GPT-4o mini
To determine if the given function \( f(x) = \frac{x^2 + 6}{x^6} \) is a polynomial, we first need to examine its form.
A polynomial function is defined as a function that can be expressed in the form:
\[
P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0
\]
where \( a_n, a_{n-1}, \ldots, a_0 \) are constants (coefficients) and \( n \) is a non-negative integer indicating the highest degree of \( x \) in the polynomial.
To analyze \( f(x) \):
1. The expression in the numerator, \( x^2 + 6 \), is a polynomial of degree 2.
2. The expression in the denominator, \( x^6 \), is not a polynomial on its own, but we can rewrite the function:
\[
f(x) = \frac{x^2 + 6}{x^6} = x^{-4} + 6x^{-6}
\]
This form shows that \( f(x) \) consists of terms with negative exponents.
Because \( f(x) \) contains terms with negative powers of \( x \), it cannot be expressed as a polynomial (which requires all exponents of \( x \) to be non-negative).
Therefore, we conclude that:
**It is not a polynomial.**
A polynomial function is defined as a function that can be expressed in the form:
\[
P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0
\]
where \( a_n, a_{n-1}, \ldots, a_0 \) are constants (coefficients) and \( n \) is a non-negative integer indicating the highest degree of \( x \) in the polynomial.
To analyze \( f(x) \):
1. The expression in the numerator, \( x^2 + 6 \), is a polynomial of degree 2.
2. The expression in the denominator, \( x^6 \), is not a polynomial on its own, but we can rewrite the function:
\[
f(x) = \frac{x^2 + 6}{x^6} = x^{-4} + 6x^{-6}
\]
This form shows that \( f(x) \) consists of terms with negative exponents.
Because \( f(x) \) contains terms with negative powers of \( x \), it cannot be expressed as a polynomial (which requires all exponents of \( x \) to be non-negative).
Therefore, we conclude that:
**It is not a polynomial.**
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