To determine which of the given expressions is a polynomial, let's carefully examine each one of them. A polynomial is an expression of the form [tex]\( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)[/tex], where each [tex]\( a_i \)[/tex] is a constant and [tex]\( n \)[/tex] is a non-negative integer.
### Option A: [tex]\( x^4 + x^{-4} + 16 \)[/tex]
In this expression, [tex]\( x^4 \)[/tex] is a term typically found in polynomials. However, [tex]\( x^{-4} \)[/tex] (which is [tex]\( \frac{1}{x^4} \)[/tex]) involves a negative exponent. Polynomials do not include negative exponents. Thus, this expression is not a polynomial.
### Option B: [tex]\( x^2 - 1 \)[/tex]
This expression involves [tex]\( x^2 \)[/tex] and [tex]\(-1\)[/tex], both of which fit the form of a polynomial where [tex]\( a_2 = 1 \)[/tex] and [tex]\( a_0 = -1 \)[/tex]. There are no negative exponents or terms in fractions. Therefore, this expression is a polynomial.
### Option C: [tex]\( \frac{x^6 - 2}{x^{-4} + 3} \)[/tex]
This is a rational expression where the numerator is [tex]\( x^6 - 2 \)[/tex] and the denominator is [tex]\( x^{-4} + 3 \)[/tex]. The term [tex]\( x^{-4} \)[/tex] (which is [tex]\( \frac{1}{x^4} \)[/tex]) involves a negative exponent in the denominator, making the overall expression not fitting the criterion for a polynomial since polynomials cannot have variable exponents in denominators or negative exponents. Therefore, this expression is not a polynomial.
### Option D: [tex]\( \frac{1}{x} + 2 \)[/tex]
In this case, [tex]\( \frac{1}{x} \)[/tex] involves a negative exponent ([tex]\( x^{-1} \)[/tex]). Additionally, polynomials cannot have variables in the denominator. This means this expression is not a polynomial.
### Conclusion
Out of the given expressions, the only one that conforms to the definition of a polynomial, with non-negative integer exponents, is option B:
[tex]\[ x^2 - 1 \][/tex]
Thus, the answer is:
B. [tex]\( x^2 - 1 \)[/tex]
